# Fuzzy-Approximation-Based Novel Back-Stepping Control of Flexible Air-Breathing Hypersonic Vehicles with Nonaffine Models

Authors: Xingge Li; Gang Li
Journal: International Journal of Aerospace Engineering
 (2019)

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## Abstract

This article investigates a novel fuzzy-approximation-based nonaffine control strategy for a flexible air-breathing hypersonic vehicle (FHV). Firstly, the nonaffine models are decomposed into an altitude subsystem and a velocity subsystem, and the nonaffine dynamics of the subsystems are processed by using low-pass filters. For the unknown functions and uncertainties in each subsystem, fuzzy approximators are used to approximate the total uncertainties, and norm estimation approach is introduced to reduce the computational complexity of the algorithm. Aiming at the saturation problem of actuator, a saturation auxiliary system is designed to transform the original control problem with input constraints into a new control problem without input constraints. Finally, the superiority of the proposed method is verified by simulation.


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## Body

## 1. Introduction

Air-breathing hypersonic vehicle (AHV) is a kind of high-velocity aircraft flying at more than 5 times the speed of sound and cruising at an altitude in the near space. With the increasing extension of human activities to space, high-velocity and high-altitude vehicles represented by air-breathing hypersonic vehicles will play an important role in the field of aerospace. The control system is the “nerve center” of an aircraft, and the design of the AHV control system has become one of the frontier issues in the field of control science [1].Compared with traditional aircrafts, the flight environment of flexible air-breathing hypersonic vehicles is more complex, the flight envelope span is larger, and the aerodynamic characteristic change is more complex. These problems bring difficulties to the design of the FHV control system. At the same time, in the control theory, FHV has the characteristics of strong nonlinearity, strong coupling, fast time-varying, and uncertainty, which make FHV extremely sensitive to flight attitude in dynamic characteristics, and the design of the FHV control system is also subject to many constraints [2–4].For the control of FHV, previous research has achieved fruitful results, for example, sliding mode variable structure control [5–8], neural network control [9–11], prescribed performance control [12–14], and fuzzy control [15, 16]. Hu et al. designed the FHV robust controller and the observers by means of sliding mode control in [6]. But in practical application, chattering problem exists in sliding mode control. In [7], the method of multimode switching between ordinary sliding surface and integral sliding surface is proposed to solve the chattering problem, but the switching between sliding surfaces is still relatively complex. In [8], Song et al. studied the double-loop sliding mode control method of HFV, unknown disturbances are observed by sliding mode observers, which enhanced the robustness of the control law. Bu et al. proposed an adaptive neural control method in [10]. The simulation results show that the control method can still guarantee the stable tracking of flight velocity and altitude to their respective reference inputs. In [11], the altitude subsystem of HFV is rewritten to a pure feedback form, and an adaptive neural control law is designed to ensure the realizability of the control law. By constructing the prescribed performance functions in [14], the velocity tracking error and altitude tracking error of FHV are guaranteed, and the prescribed transient performance and steady-state accuracy of the system are satisfied. In [16], unknown functions of FHV models are approximated online by using fuzzy systems, but the algorithm requires that the unknown functions be strictly positive and bounded. There are a lot of unknown functions in each subsystem, and computational costs are heavy.Most of the above studies focus on the affine models of FHV, which require simplifying the nonaffine models of FHV to the complete affine forms of the control input under certain assumptions, and then conducting research [17]. However, in fact, many of the dynamic performances of FHV are nonlinear functions of rudder deflection, angle of attack, velocity, and altitude. Particularly in the hypersonic flight process of FHV, these nonaffine dynamic characteristics cannot be affine, so it is necessary to strengthen the study of nonaffine models of FHV. [18] designed a control law of FHV based on the Mean Value Theorem [19], which has achieved certain effects. In [20], for a class of high-order nonaffine nonlinear systems with completely unknown dynamics, Meng et al. developed a novel system transformation to transform the nonaffine system into an affine system. According to the idea expressed in [20, 21] combined the system transformation with prescribed performance control scheme, the simulation results verified the effectiveness of the method. In [22], the novel system transformation with a low-pass filter is applied to constrained systems.Based on above research, a fuzzy-approximation-based novel back-stepping control scheme of FHV with nonaffine models is proposed in this article. The FHV’s nonaffine dynamics is decomposed into velocity subsystem and altitude subsystem to be controlled separately, and then nonaffine dynamics of FHV is processed by low-pass filters. A fuzzy approximator is introduced to approximate unknown function in each subsystem, and norm estimation approach [23] is introduced in the approximation of unknown functions. A novel back-stepping control method without virtual control law is designed. Considering the constraints of actuators, a new auxiliary system is designed to transform the control constraints into unconstrained ones. Finally, based on the simulation experiments, the performance of the control method is verified. The special contributions of this article are summarized as follows:
(1)
The traditional affine models need to simplify the nonaffine dynamics into fully affine forms. In this article, low-pass filters are used to solve the nonaffine dynamics of FHV
(2)
For each subsystem of FHV, only a fuzzy approximator is introduced to approximate the total uncertainties in the subsystem, which reduces the computational complexity of the system on the basis of ensuring the robustness of the controller. At the same time, the norm estimation approach is introduced in this article to ensure the computational speed compared with the previous fuzzy control method
(3)
Compared with the traditional back-stepping control method, all the virtual controllers proposed in this article are only used as intermediate variables for the stability analysis of the system and do not participate in the final solution and execution
(4)
An auxiliary system is introduced to transform the original control problem with input constraints into a new control problem without input constraints to solve the transient saturation problem in the control input


## 2. FHV Model and Preliminaries

### 2.1. FHV Model

Taking X-43A as the modeling objective, the longitudinal integration analytical model of FHV was established by Bolender and Doman [24]; Parker et al. ignored the weak coupling in the dynamic model and established the parameter fitting model of FHV [17].
(1)V̇=Tcosθ−γ−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+Tsinθ−γmV−gcosγV,(4)θ̇=Q,(5)Q̇=M+Ψ~1η¨1+Ψ~2η¨2Iyy,(6)k1η¨1=−2ζ1ω1η̇1−ω12η1+N1−Ψ~1MIyy−Ψ~1Ψ~2η¨2Iyy,(7)k2η¨2=−2ζ2ω2η̇2−ω22η2+N2−Ψ~2MIyy−Ψ~2Ψ~1η¨1Iyy,where
(8)k1=1+Ψ~1Iyy,k2=1+Ψ~2Iyy,Ψ~1=∫−Lf0m^fξφfξdξ,Ψ~2=∫0Lam^aξφaξdξ,where φf∙ and φa∙ are structural mode shapes; other variables and functions are given in [24].The FHV models contain five rigid body states and two elastic modes. The five rigid body states are flight velocityV, flight altitude h, track angle γ, pitching angle θ, and pitch rate Q; η1 and η2 represent the two elastic modes of FHV; the distance between the vehicle and the earth’s center is expressed by r; m is the mass of FHV; moment of inertia is Iyy; define the flight angle of attack α=θ−γ. ς1 and ς2 represent the damping ratio, ω1 and ω2 represent the natural frequency for elastic modes, respectively; N1 and N2 are generalized forces; T, D, L, and M represent the thrust, resistance, lift, and pitching moment of FHV, respectively; where the approximations of T, D, L, M, N1, and N2 are designed as follows [17]:
(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,q¯=12ρV2,ρ=ρ0exp−h−h0hs,with
(10)CTα3=β1h,q¯Φ+β2h,q¯,CTα2=β3h,q¯Φ+β4h,q¯,CTα=β5h,q¯Φ+β6h,q¯,CT0=β7h,q¯Φ+β8h,q¯.The control inputs are fuel-air ratioΦ and elevator deflection angle δe; the average aerodynamic chord length of FHV is c¯; the reference area is S; q¯ represents the dynamic pressure of FHV; the air density at h is ρ; zT is the coupling coefficient of thrust and moment; ce represents rudder deflection coefficient; h0 represents nominal altitude; the air density at nominal altitude is ρ0; 1/hs is the decay rate of air density.Remark 1.
In the following controller design, the rigid body states can be measured, and the elastic modes are usually regarded as the unknown dynamics, which are treated as the robustness of the controller [10, 25].Remark 2.
It can be seen from equation (10) that the FHV models are nonaffine for the control inputs, because the parameter fitting form of resistance D contains δe2 [17]. Therefore, in the subsequent control system design, based on the back-stepping control framework, in the case of uncertainties and nonaffine dynamics, fuzzy-approximation-based nonlinear tracking control laws Φ and δe are designed to achieve robust tracking of velocity and altitude commands Vref and href.Remark 3.
Because the fuel-air ratioΦ is related to thrust T, so the velocity V is mainly controlled by Φ. The elevator angle δe controls the change of altitude h by affecting the pitch rate Q, pitch angle θ, and track angle γ. Therefore, the FHV models can be decomposed into velocity subsystem (equation (1)) and altitude subsystem (equations (2)-(5)), and then the control laws are designed separately [26].


### 2.2. Theory of Fuzzy Approximation

In recent years, the universal approximation characteristics of the fuzzy system have been gradually discovered, proposed, and applied. Under sufficient rules of the fuzzy system, the fuzzy system can approximate the given nonlinear continuous function with arbitrary precision [27, 28]. The complete fuzzy system includes inference engine, knowledge base, fuzzification interface, and defuzzification interface. Compared with neural network and polynomial function approximation, the advantage of the fuzzy system is that it can effectively control based on fuzzy logic. Where the input of the system X=x1,x2,⋯,xnT∈Rn, the output is F^X.
(11)F^X∣θ=θ∗TξX.In the fuzzy system,θ∗=θ1∗,θ2∗,⋯,θp∗T∈Rp represents the parameter vector of ideal weight coefficient, fuzzy basis function ξX=ξ1X,ξ2X,⋯,ξp,XT∈Rp. Fuzzy basis functions are expressed as
(12)ξjX=∏i=1nμFijxi∑l=1N∏i=1nμFijxi.In equation (12), Fij is the jth j=1,⋯,N fuzzy rule corresponding to the ith i=1,⋯,n input point, N is the number of fuzzy rules, μFij is the membership function, 0≤μFij≤1. In this article, the Gauss basis function is chosen as the membership function.Lemma 1 [27].
Suppose that the real continuous functionFX is an arbitrary function defined on compact set Ω. For any μ>0, the fuzzy system F^X=θ∗TξX can be obtained, and the following conditions are guaranteed:
(13)supx∈ΩFX−θ∗TξX≤μ.


### 2.3. Saturation Auxiliary System

In order to maintain the normal working mode of scramjet and the physical constraint of FHV by deflection angle, the actuators of control inputsΦ and δe are limited to some extent. The saturation problem of control input is described as follows:
(14)Φc=Φmax,Φ>Φmax,Φ,Φmin≤Φ≤Φmax,Φmin,Φ<Φmin,(15)δec=δemax,δe>δemax,δe,δemin≤δe≤δemin,δe<δemin,δemax,where Φc denotes the fuel-air ratio in the case of input saturation, δec denotes the elevator deflection in the case of input saturation, Φmin and Φmax are the lower and upper bounds of Φ, and δemin and δemax are the lower and upper bounds of δe. In this article, a saturation auxiliary system is designed under the condition of input saturation. The input constraints of the control system are transformed into unconstrained ones.When the inputs of the saturated auxiliary system areΦ and δe, the sigmoid functions are used to approximate the outputs Φc and δec of the auxiliary system.
(16)Φc=Φmax−Φmin2sigmoid2Φ−Φmax+ΦminΦmax−Φmin+Φmax+Φmin2,δec=δemax−δemin2sigmoid2δe−δemin+δemaxδemax−δemin+δemax+δemin2.According to equation (16) and the characteristic of the sigmoid function, whether Φ and δe take any value, the saturation auxiliary system can make Φc and δec meet the constraints of equation (15).Remark 4.
In the controller design, the original control objective can be converted to design unconstrainedΦ and δe such that velocity Vand altitude h track their reference commands Vref and href.


### 2.4. Controller Design

In the following section, the altitude subsystem and velocity subsystem of the controller are designed with the new fuzzy-approximation-based control scheme in this article.


#### 2.4.1. Controller Design of the Altitude Subsystem

In the altitude subsystem, we design an altitude controller based on nonaffine dynamics and a fuzzy approximator to achieve stable tracking of reference altitudehref.Define altitude tracking errorh~=h−href, the reference command of γ is chosen as
(17)γd=arcsin−kh,1h~−kh,2∫0th~dτ+ḣrefV,where kh,1∈R+ and kh,2∈R+ are design parameters.If the design parameterskh,1>0, kh,1>0 and γ→γd, then h~ can be regulated to zero exponentially [29]. In this way, the control objective of the altitude subsystem can be transformed to γ→γd by designing the appropriate control law δe.According to the idea expressed in [20, 21], we define a set of newly state variables; the control input δe is also viewed as the newly defined state to deal with the nonaffine features. Define new state variables
(18)ψh,1=γ,ψh,2=θ,ψh,3=Q,ψh,4=δe.Original nonaffine systems (3)-(5) can be rewritten to the following equivalent general nonaffine form according to equation (18).
(19)ψ̇h,1=Ж1ψh,1,ψh,2,ψ̇h,2=ψh,3,ψ̇h,3=Ж3ψh,1,ψh,2,ψh,3,ψh,4,ψ̇h,4=uh−τhψh,4,yh=ψh,1=γ,where Ж1ψh,1,ψh,2 and Ж3ψh,1,ψh,2,ψh,3,ψh,4 are continuous differentiable nonlinear functions; uh is an introduced variable, and it is regarded as the virtual control input of (19); the control inputs δe are generated by a low-pass filter “ψ̇h,4=uh−τhψh,4” driven by uh. “ψ̇h,4=uh−τhψh,4” is input-to-state stable. yh is the output of (19); τh∈R+is the design parameter.From equations (3)-(5) together with previous studies [17], we have
(20)∂Ж1ψh,1,ψh,2∂ψh,1≈q¯SCLα+Tcosθ−γmV>0,∂Ж3ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4≈q¯Sc¯ce+ς~2N2δe/k2Iyy>0.Next, transform the nonaffine models.Defineυ1=ψh,1=γ, υ2=υ̇1=Ж1ψh,1,ψh,2, combined with equation (19), the time derivative of υ2⁠ is derived as
(21)υ̇2=∂Ж1ψh,1,ψh,2∂ψh,1ψ̇h,1+∂Ж1ψh,1,ψh,2∂ψh,2ψ̇h,2=∂Ж1ψh,1,ψh,2∂ψh,1Ж1ψh,1,ψh,2+∂Ж1ψh,1,ψh,2∂ψh,2ψh,3≜ϒ1ψh,1,ψh,2,ψh,3.It can be noted that(22)∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3=∂Ж1ψh,1,ψh,2∂ψh,2>0.We further defineυ3=υ̇2=Y1ψh,1,ψh,2,ψh,3. Taking time derivative of υ3 and substituting (19)
(23)υ̇3=∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,1ψ̇h,1+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,2ψ̇h,2+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3ψ̇h,3=∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,1Ж1ψh,1,ψh,2+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,2ψh,3+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3Ж3ψh,1,ψh,2,ψh,3,ψh,4≜ϒ2ψh,1,ψh,2,ψh,3,ψh,4.Finally, defineυ4=υ̇3=Y2ψh,1,ψh,2,ψh,3,ψh,4. Invoking (19), the time derivative of υ4 is derived as
(24)υ̇4=∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,1ψ̇h,1+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,2ψ̇h,2+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,3ψ̇h,3+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4ψ̇h,4=∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,1Ж1ψh,1,ψh,2+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,2ψh,3+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,3Ж3ψh,1,ψh,2,ψh,3,ψh,4−∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4τhψh,4+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4uh≜ϒ3ψh,1,ψh,2,ψh,3,ψh,4+Khuh.where
(25)Κh=∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4=∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3∂Ж3ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4>0.Through model transformation, the nonaffine system (19) can be converted into affine form.
(26)υ̇1=υ2,υ̇2=υ3,υ̇3=υ4,υ̇4=Y3ψh,1,ψh,2,ψh,3,ψh,4+Κhuh,yh=υ1=γ.The fuzzy-approximation-based altitude subsystem controller is designed by using the transformed affine form.Remark 5.
According to the theory of input-to-state stability, when the input is bounded, the state of the system is also bounded. So we need to prove the boundedness ofuh.Theorem 1.
If the control inputsuh are bounded, there must be fixed values uhM, such that
(27)uh≤uhM.Proof.
Define the Lyapunov function(28)Lψh,4=ψh,422.
Taking time derivative along (28).
(29)L̇ψh,4=ψh,4ψ̇h,4=ψh,4uh−τhψh,4=−τhψh,42+uhψh,4≤−τhψh,42+uhM22+uh,422=−τh−12uh,422+uhM22
Ifτh>1/2, ψh,4 will ultimately converge to the compact Ωψh,4, where Ωψh,4=ψh,4∣ψh,4≤uhM2/2/τh−1/2. That is, ψh,4 is bounded and the low-pass filter “ψ̇h,4=uh−τhψh,4” is stable when uh is bounded.
Define tracking erroreh,1 as
(30)eh,1=υ1−υ1d.
Taking time derivative along (30) and employing (26), it leads to
(31)ėh,1=υ̇1−υ̇1d=υ2−υ̇1d.
Define intermediate variableυ2d as
(32)υ2d=−λh,1eh,1+υ̇1d,where λh,1∈R+ is the design parameter.
Then, define tracking erroreh,2 as
(33)eh,2=υ2−υ2d.
Applying (26), the time derivative of eh,2 is derived as
(34)ėh,2=υ̇2−υ̇2d=υ3−υ̇2d.
Define a new intermediate state variableυ3d as
(35)υ3d=−λh,2eh,2−eh,1+υ̇2d,where λh,2∈R+ is the design parameter.
Define tracking erroreh,3 as
(36)eh,3=υ3−υ3d.
Differentiating along (36) with respect to time and applying (26), we get
(37)ėh,3=υ̇3−υ̇3d=υ4−υ̇3d.
Then, the intermediate variableυ4d is designed as
(38)υ4d=−λh,3eh,3−eh,2+υ̇3d,where λh,3∈R+ is the design parameter.
Finally, define tracking erroreh,4 as
(39)eh,4=υ4−υ4d.
Taking the time derivative ofeh,4 and invoking (26) lead to
(40)ėh,4=υ̇4−υ̇4d=Y3ψh,1,ψh,2,ψh,3,ψh,4+Κhuh−υ̇4d=ΚhY3ψh,1,ψh,2,ψh,3,ψh,4−υ̇4dΚh+uh.
Define unknown functionFh as
(41)Fh=Y3ψh,1,ψh,2,ψh,3,ψh,4−υ̇4dΚh+uh=FhΨh,where Ψh=ψh,1,ψh,2,ψh,3,ψh,4T. Since the function Fh is an unknown function, the fuzzy function approximator designed in the above part is used to approximate it. The estimated value F^h of the nonlinear function Fh can be expressed as
(42)F^hΨh∣θh=θh∗TξhΨh,where Ψh=ψh,1,ψh,2,ψh,3,ψh,4T∈R+ is the input vector to the unknown function, fuzzy basis function ξhΨh=ξh,1Ψh,ξh,2Ψh,⋯,ξh,p1,ΨhT∈Rp1, θh∗=θh,1∗,θh,2∗,⋯,θh,p1∗T∈Rp1is the ideal weight coefficient parameter vector, and the unknown function FV can be expressed as
(43)FhΨh=θh∗TξhΨh+μh,where μh represents the estimated error, and μh≤μhM; μhM is the upper bound of the estimated error.
Finally, the control law is devised as(44)uh=−λh,4eh,4−eh,3−φ^12ξhTΨhξhΨh,where λh,4∈R+ is the design parameter. φ^1 denotes the estimation of φ1=θh∗2; its adaptive law is selected as
(45)φ^̇1=l12eh,1ξhTΨhξhΨh−2φ^1,where l1∈R+ is the design parameter.
Substitute equation (32) into equation (33) to obtain
(46)eh,2=υ2−−λh,1eh,1+υ̇1d=υ2+λh,1eh,1−υ̇1d=ėh,1+λh,1eh,1.
Taking time derivative along (46),
(47)ėh,2=e¨h,1+λh,1ėh,1.
By combining equations (26) and (35), equation (36) can be obtained.
(48)eh,3=υ3−−λh,2eh,2−eh,1+υ̇2d=υ3+λh,2eh,2+eh,1−υ̇2d=ėh,2+λh,2eh,2+eh,1.
Substituting equations (46) and (47) into equation (48), it yields
(49)eh,3=e¨h,1+λh,1ėh,1+λh,2ėh,1+λh,1eh,1+eh,1=e¨h,1+λh,1+λh,2ėh,1+λh,1λh,2+1eh,1.
Taking time derivative along (49), we obtain
(50)ėh,3=e⃛h,1λh,1+λh,2e¨h,1+λh,1λh,2+1ėh,1.
Substituting (26) and (38) into (39), eh,4 is formulated as
(51)eh,4=υ4−−λh,3eh,3−eh,2+υ̇3d=υ4+λh,3eh,3+eh,2−υ̇3d=ėh,3+λh,3eh,3+eh,2.
Combining equations (46), (49), and (50), we can obtain
(52)eh,4=e⃛h,1+λh,1+λh,2e¨h,1+λh,1λh,2+1ėh,1+λh,3e¨h,1+λh,1+λh,2ėh,1+λh,1λh,2+1eh,1+ėh,1+λh,1eh,1=e⃛h,1+λh,1+λh,2+λh,3e¨h,1+λh,1λh,2+λh,1λh,3+λh,2λh,3+2ėh,1+λh,1+λh,3+λh,1λh,2λh,3eh,1.
Substituting (49) and (52) into (44) leads to
(53)uh=−λh,4e⃛h,1−λh,4λh,1+λh,4λh,2+λh,4λh,3+1e¨h,1−λh,4λh,1λh,2+λh,4λh,2λh,3+λh,4λh,3λh,1+2λh,4+λh,1+λh,2ėh,1−λh,4λh,1λh,2λh,3+λh,4λh,3+λh,4λh,1+λh,1λh,2+1eh,1−φ^12ξhTΨhξhΨh.Remark 6.
Unlike the back-stepping control method proposed by Bu et al. [30], the presented altitude controller is exploited using an improved back-stepping framework, in which the virtual controller is transformed into intermediate variables and only the final actual control laws are needed to be implemented, which avoids repeated calculation and greatly reduces the computational pressure of the system.Remark 7.
Compared with the method in [16, 31, 32], each weight update requires a lot of calculation. In this article, only one online learning parameter is needed by using the norm estimation approach, which makes the online calculation of the system significantly reduced.Theorem 2.
Considering the closed-loop subsystem consisting of plant (26), with controller (53), and adaptive law (45). Then all the signals involved are semiglobally uniformly ultimately bounded.Proof.
Consider the following Lyapunov function:(54)Wh=eh,122+eh,222+eh,322+eh,422Κh+φ~122l1with φ~1=φ^1−φ1.
Applying (31), (34), (37), (40), (41), and (45), Ẇhis derived as
(55)Ẇh=eh,1ėh,1+eh,2ėh,2+eh,3ėh,3+eh,4ėh,4Κh+φ~1φ^̇1l1=eh,1υ2−υ̇1d+eh,2υ3−υ̇2d+eh,3υ4−υ̇3d+eh,4Y3ψh,1,ψh,2,ψh,3,ψh,4−υ̇4dΚh+uh+φ~1l1l12eh,1φ^1ξhTΨhξhΨh−2φ^1=eh,1υ2d+eh,2−υ̇1d+eh,2υ3d+eh,3−υ̇2d+eh,3υ4d+eh,4−υ̇3d+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+μh+uh−2φ~1φ^1l1.
Considering (32), (35), and (38), Ẇhbecomes
(56)Ẇh=eh,1−λh,1eh,1+υ̇1d+eh,2−υ̇1d+eh,2−λh,2eh,2−eh,1+υ̇2d+eh,3−υ̇2d+eh,3−λh,3eh,3−eh,2+eh,4+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+μh+uh−2φ~1φ^1l1.
Substituting (44) into (56) yields, we can get
(57)Ẇh=−λh,1eh,12−λh,2eh,22−λh,3eh,32+eh,3eh,4+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+eh,4μh+eh,4−λh,4eh,4−eh,3−φ^12ξhTΨhξhΨh−2φ~1φ^1l1=−λh,1eh,12−λh,2eh,22−λh,3eh,32−λh,4eh,42+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh−eh,4φ^12ξhTΨhξhΨh+eh,4μh−2φ~1φ^1l1.
From the fact that(58)φ~12+2φ~1φ^1−φ~1+φ12=φ~12+2φ~1φ1+φ12=φ~1+φ12≥0,
we obtain(59)2φ~1φ^1≥φ~12−φ12.
Thereby, equation (57) becomes
(60)Ẇh≤−λh,1eh,12−λh,2eh,22−λh,3eh,32−λh,4eh,42+12eh,1φ~1ξhTΨhξhΨh+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+eh,4μh−eh,4φ~12ξhTΨhξhΨh−eh,4φ12ξhTΨhξhΨh−φ~12l1+φ12l1.
In this article, since0≤μFijx≤1, and applying (12), we know that 0≤ξjX≤1. There exists a constantξ¯h∈R+, such that ξhΨh≤ξ¯h. Hence, we get
(61)12eh,1φ~1ξhTΨhξhΨh≤12ξ¯h2eh,1φ~1≤2l1212ξ¯h2eh,12+φ~1222l1=l1ξ¯h44eh,12+φ~124l1,eh,4θh∗TξhΨh≤eh,4φ1ξ¯h≤φ1ξ¯heh,422+12=φ1ξ¯h2eh,42+φ1ξ¯h2,eh,4μh≤μhMeh,4≤μhMeh,422+12=μhMeh,422+μhM2−eh,4φ~12ξhTΨhξhΨh≤12ξ¯h2eh,4φ~1≤2l1212ξ¯h2eh,42+φ~1222l1=l1ξ¯h44eh,42+φ~124l1−eh,4φ12ξhTΨhξhΨh≤12ξ¯h2eh,4φ1≤12φ1ξ¯h2eh,422+12=φ1ξ¯h24eh,42+φ1ξ¯h24.
Then, (60) becomes
(62)Ẇh≤−λh,1eh,12−λh,2eh,22−λh,3eh,32−λh,4eh,42+l1ξ¯h44eh,12+φ~124l1+φ1ξ¯h2eh,42+φ1ξ¯h2+μhMeh,422+μhM2+l1ξ¯h44eh,42+φ~124l1+φ1ξ¯h24eh,42+φ1ξ¯h24−φ~12l1+φ12l1,Ẇh≤−λh,1−l1ξ¯h44eh,12−λh,2eh,22−λh,3eh,32+φ1ξ¯h2+μhM2+φ1ξ¯h24−λh,4−φ1ξ¯h2−μhM2−l1ξ¯h44−φ1ξ¯h24eh,42+φ12l1−φ~122l1.
Letλh,1>l1ξ¯h4/4, λh,4>φ1ξ¯h/2+μhM/2+l1ξ¯h4/4+φ1ξ¯h2/4, λh,2>0, λh,3>0, and l1>0. And define the following compact sets
(63)Ωeh,1=eh,1∣eh,1≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,1−l1ξ¯h44,Ωeh,4=eh,4∣eh,4≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,4−φ1ξ¯h2−μhM2−l1ξ¯h44−φ1ξ¯h24,Ωeh,2=eh,2∣eh,2≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,2,Ωeh,3=eh,3∣eh,3≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,3,Ωφ~1=φ~1∣φ~1≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/12l1.
Ifeh,1∉Ωeh,1 or eh,4∉Ωeh,4 or eh,2∉Ωeh,2 or eh,3∉Ωeh,3 or φ~1∉Ωφ~1, Ẇh will be negative. It is clear that eh,1, eh,4, eh,2, eh,3 and φ~1 are semiglobally uniformly ultimately bounded. This completes the proof.


### 2.5. Controller Design of Velocity Subsystem

For the velocity subsystem, its tracking target is to achieve the stable tracking of velocityV to reference velocity Vref.Define new state variablesψV,1=VandψV,2=Φ. In order to ensure the control effect, the velocity subsystem should be changed into an equivalent general nonaffine form with ψV,1 and ψV,2.
(64)ψ̇V,1=ΓψV,1,ψV,2,ψ̇V,2=uV−τVψV,2,yV=ψV,1,where the transformation consists of a low-pass filter “ψ̇V,2=uV−τVψV,2” and 2 newly defined state variables. ΓψV,1,ψV,2 is an unknown continuous differentiable nonlinear function, uV is the input of (64), and yV is the output of (64); τV∈R+ is the design parameter.Then, define state variablesε1 and ε2. Where ε1=yV, ε2=ε̇1=ψ̇V,1=ΓψV,1,ψV,2, we can get
(65)ε̇2=∂ΓψV,1,ψV,2∂ψV,1ψ̇V,1+∂ΓψV,1,ψV,2∂ψV,2ψ̇V,2=∂ΓψV,1,ψV,2∂ψV,1ΓψV,1,ψV,2+∂ΓψV,1,ψV,2∂ψV,2uV−τVψV,2=∂ΓψV,1,ψV,2∂ψV,1ΓψV,1,ψV,2+∂ΓψV,1,ψV,2∂ψV,2uV−τV∂ΓψV,1,ψV,2∂ψV,2ψV,2≜YVψV,1,ψV,2=∂ΓψV,1,ψV,2∂ψV,1ΓψV,1,ψV,2+ΚVuV,Where nonlinear continuous functions YVψV,1,ψV,2 and КV are expressed as
(66)YVψV,1,ψV,2=∂ΓψV,1,ψV,2∂ψV,1ΓψV,1,ψV,2τV∂ΓψV,1,ψV,2∂ψV,2ψV,2,ΚV=∂ΓψV,1,ψV,2∂ψV,2=∂Tcosα−D/m−ginγ∂Φ=cosαmβ1h,q¯α3+β3h,q¯α2+β5h,q¯α+β7h,q¯.Remark 8.
Noting thatα∈−8.7rad,8.7rad,×10−2 α∈−8.7rad,8.7rad×10−2, so we can get the definitions of α2 and α3 as α2∈0rad2,7.57rad2×10−3 and α3∈−6.5850rad3,6.5850rad3×10−4, respectively. And we can calculate that КV>0.According to the above analysis, we can express the nonaffine model in the form of the affine model, as shown in equation (67).
(67)ε̇1=ε2,ε̇2=YVψV,1,ψV,2+ΚVuV,yV=ψV,1.Next, we need to prove the boundedness ofuV.Theorem 3.
If the control inputuV is bounded, there must be a fixed value uVM, such that
(68)uV≤uVM.Proof.
Construct the Lyapunov function(69)LψV,2=ψV,222.
Taking time derivative along (69)
(70)L̇ψV,2=ψV,2ψ̇V,2=ψV,2uV−τVψV,2=−τVψV,22+uVψV,2≤−τVψV,22+uVM22+ψV,222=−τV−12ψV,222+uVM22.
IfτV>1/2,ψV,2 will ultimately converge to the compact ΩψV,2. Where ΩψV,2=ψV,2∣ψV,2≤uVM2/2/τV−1/2. As a result, the boundedness of uh leads to the newly introduced system “ψ̇V,2=uV−τVψV,2” which is stable.
Define tracking erroreV,1 as
(71)eV,1=ε1−ε1d=ε1−Vref.
Taking time derivative along (71) and employing (67), it leads to
(72)ėV,1=ε̇1−ε̇1d=ε2−ε̇1d.
Define intermediate variableε2d as
(73)ε2d=−λV,1eV,1+ε̇1d,where λV,1∈R+ is the design parameter.
Define tracking erroreV,2 as
(74)eV,2=ε2−ε2d.
Utilizing (67) and (73), the time derivative of eV,2 is derived as
(75)ėV,2=ε̇2−ε̇2d=YVψV,1,ψV,2+ΚVuV−ε̇2d=ΚVYVψV,1,ψV,2−ε̇2dΚV+uV≜ΚVFVΨV,where ΨV=ψV,1,ψV,2T. Since the function FV is an unknown function, the fuzzy function approximator is used to approximate it. The estimated value F^V of the nonlinear function FV can be expressed as
(76)F^VΨV∣θV=θV∗TξVΨV.
ΨV=ψV,1,ψV,2T∈R+ is the input vector of the unknown function; θV∗=θV,1∗,θV,2∗,⋯,θV,p2∗T∈Rp2 is the parameter vector of ideal weight coefficient, fuzzy basis function ξVΨV=ξV,1ΨV,ξV,2ΨV,⋯,ξV,p2,ΨVT∈Rp2. The unknown function FV can be expressed as
(77)FVΨV=θV∗TξVΨV+μV,where μV stands for estimation error, μV≤μVM; μVM represents the upper bound of estimation error.
The velocity control law is designed as(78)uV=−λV,2eV,2−eV,1−φ^22ξVTΨVξVΨV,where φ^2=θV∗2, and its adaptive law is chosen to be
(79)φ^̇2=l22eV,1ξVTΨVξVΨV−2φ^2,where l2∈R+ is the design parameter.
Substituting equations (67) and (73) into equation (74), we can obtain
(80)eV,2=ε2−ε2d=ε̇1−ε2d=ε̇1−−λV,1eV,1+ε̇1d=ε̇1−ε̇1d+λV,1eV,1=eV,1+λV,1eV,1.
Substitute equation (80) into equation (78) to get
(81)uV=−λV,2eV,1+λV,1eV,1−eV,1−φ^22ξVTΨVξVΨV=−λV,2ėV,1−λV,1λV,2eV,1−φ^22ξVTΨVξVΨV.Theorem 4.
Considering the closed-loop subsystem consisting of plant (67), with controller (81) and adaptive law (79). Then, all the signals involved are semiglobally uniformly ultimately bounded.Proof.
Consider the following Lyapunov function candidate:(82)WV=eV,122+eV,222ΚV+φ~222l2with φ~2=φ^2−φ2.
Applying (73), (74) into (72), ẆV is expressed as
(83)ẆV=eV,1ėV,1+eV,2ėV,2ΚV+φ~2φ^̇2l2=eV,1eV,2−λV,1eV,1+eV,2ėV,2ΚV+φ~2φ^̇2l2.
Considering (75), (77), and (79), ẆV becomes
(84)ẆV=eV,1eV,2−λV,1eV,1+eV,2YVψV,1,ψV,2−ε̇2dΚV+uV+φ~2φ^̇2l2=eV,1eV,2−λV,1eV,12+eV,2θV∗TξVΨV+μV+uV+12eV,1φ~2ξVTΨVξVΨV−2φ~2φ^2l2.
To simplify equation (84) and substituting (78) into (84) yields, we can get
(85)ẆV=eV,1eV,2−λV,1eV,12+eV,2θV∗TξVΨV+μV−λV,2eV,2−eV,1−φ^22ξVTΨVξVΨV+12eV,1φ~2ξVTΨVξVΨV−2φ~2φ^2l2=−λV,1eV,12−λV,2eV,22+12eV,1φ~2ξVTΨVξVΨV+eV,2θV∗TξVΨV+eV,2μV−eV,2φ~22ξVTΨVξVΨV−eV,2φ22ξVTΨVξVΨV−2φ~2φ^2l2.
From the fact that(86)φ~22+2φ~2φ^2−φ~2+φ22=φ~22+2φ~2φ2+φ22=φ~2+φ22≥0,
we obtain(87)2φ~2φ^2≥φ~22−φ22.
Thereby, equation (85) becomes
(88)ẆV≤−λV,1eV,12−λV,2eV,22+12eV,1φ~2ξVTΨVξVΨV+eV,2θV∗TξVΨV−eV,2φ~22ξVTΨVξVΨV−eV,2φ22ξVTΨVξVΨV+eV,2μV−φ~22l2+φ22l2.
In this article, since0≤μFijx≤1, and applying (12), we know that 0≤ξjX≤1.There exists a constantξ¯V∈R+, such that ξVΨV≤ξ¯V. Hence, we get
(89)12eV,1φ~2ξVTΨVξVΨV≤12ξ¯V2eV,1φ~2≤2l2212ξ¯V2eV,12+φ~2222l2=l2ξ¯V44eV,12+φ~224l2,eV,2θV∗TξVΨV≤eV,2φ2ξ¯V≤φ2ξ¯VeV,222+12=φ2ξ¯V2eV,22+φ2ξ¯V2,eV,2μV≤μhMeV,2≤μVMeV,222+12=μVMeV,222+μVM2−eV,2φ~22ξVTΨVξVΨV≤12ξ¯V2eV,2φ~2≤2l2212ξ¯V2eV,22+φ~2222l2=l2ξ¯V44eV,22+φ~224l2−eV,2φ22ξVTΨVξVΨV≤12ξ¯V2eV,2φ2≤12φ2ξ¯V2eV,222+12=φ2ξ¯V24eV,22+φ2ξ¯V24.
Then, (88) becomes
(90)ẆV≤−λV,1eV,12−λV,2eV,22+l2ξ¯V44eV,12+φ~224l2+φ2ξ¯V2eV,22+φ2ξ¯V2+μVMeV,222+μVM2+l2ξ¯V44eV,22+φ~224l2+φ2ξ¯V24eV,22+φ2ξ¯V24−φ~22l2+φ22l2,ẆV≤−λV,1−l2ξ¯V44eV,12−λV,2−φ2ξ¯V2−μVM2−l2ξ¯V44−φ2ξ¯V24eV,22+φ2ξ¯V2+μVM2+φ2ξ¯V24+φ22l2−φ~222l2.
LetλV,1>l2ξ¯V4/4, λV,2>φ2ξ¯V/2+μVM/2+l2ξ¯V4/4+φ2ξ¯V2/4, l2>0. And define the following compact sets:
(91)ΩeV,1=eV,1∣eV,1≤φ2ξ¯V2+μVM2+φ22l2+φ2ξ¯V24/λV,1−l2ξ¯V44,ΩeV,2=eV,2∣eV,2≤φ2ξ¯V2+μVM2+φ22l2+φ2ξ¯V24/λV,2−φ2ξ¯V2−μVM2−l2ξ¯V44−φ2ξ¯V24,Ωφ~2=φ~2∣φ~2≤φ2ξ¯V2+μVM2+φ22l2+φ2ξ¯V24/12l2.
IfeV,1∉ΩeV,1 or eV,2∉ΩeV,2 or φ~2∉Ωφ~2, ẆV will be negative. Thus, eV,1, eV,2, and φ~2 are semiglobally uniformly ultimately bounded. This is the end of the proof.


## 2.1. FHV Model

Taking X-43A as the modeling objective, the longitudinal integration analytical model of FHV was established by Bolender and Doman [24]; Parker et al. ignored the weak coupling in the dynamic model and established the parameter fitting model of FHV [17].
(1)V̇=Tcosθ−γ−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+Tsinθ−γmV−gcosγV,(4)θ̇=Q,(5)Q̇=M+Ψ~1η¨1+Ψ~2η¨2Iyy,(6)k1η¨1=−2ζ1ω1η̇1−ω12η1+N1−Ψ~1MIyy−Ψ~1Ψ~2η¨2Iyy,(7)k2η¨2=−2ζ2ω2η̇2−ω22η2+N2−Ψ~2MIyy−Ψ~2Ψ~1η¨1Iyy,where
(8)k1=1+Ψ~1Iyy,k2=1+Ψ~2Iyy,Ψ~1=∫−Lf0m^fξφfξdξ,Ψ~2=∫0Lam^aξφaξdξ,where φf∙ and φa∙ are structural mode shapes; other variables and functions are given in [24].The FHV models contain five rigid body states and two elastic modes. The five rigid body states are flight velocityV, flight altitude h, track angle γ, pitching angle θ, and pitch rate Q; η1 and η2 represent the two elastic modes of FHV; the distance between the vehicle and the earth’s center is expressed by r; m is the mass of FHV; moment of inertia is Iyy; define the flight angle of attack α=θ−γ. ς1 and ς2 represent the damping ratio, ω1 and ω2 represent the natural frequency for elastic modes, respectively; N1 and N2 are generalized forces; T, D, L, and M represent the thrust, resistance, lift, and pitching moment of FHV, respectively; where the approximations of T, D, L, M, N1, and N2 are designed as follows [17]:
(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,q¯=12ρV2,ρ=ρ0exp−h−h0hs,with
(10)CTα3=β1h,q¯Φ+β2h,q¯,CTα2=β3h,q¯Φ+β4h,q¯,CTα=β5h,q¯Φ+β6h,q¯,CT0=β7h,q¯Φ+β8h,q¯.The control inputs are fuel-air ratioΦ and elevator deflection angle δe; the average aerodynamic chord length of FHV is c¯; the reference area is S; q¯ represents the dynamic pressure of FHV; the air density at h is ρ; zT is the coupling coefficient of thrust and moment; ce represents rudder deflection coefficient; h0 represents nominal altitude; the air density at nominal altitude is ρ0; 1/hs is the decay rate of air density.Remark 1.
In the following controller design, the rigid body states can be measured, and the elastic modes are usually regarded as the unknown dynamics, which are treated as the robustness of the controller [10, 25].Remark 2.
It can be seen from equation (10) that the FHV models are nonaffine for the control inputs, because the parameter fitting form of resistance D contains δe2 [17]. Therefore, in the subsequent control system design, based on the back-stepping control framework, in the case of uncertainties and nonaffine dynamics, fuzzy-approximation-based nonlinear tracking control laws Φ and δe are designed to achieve robust tracking of velocity and altitude commands Vref and href.Remark 3.
Because the fuel-air ratioΦ is related to thrust T, so the velocity V is mainly controlled by Φ. The elevator angle δe controls the change of altitude h by affecting the pitch rate Q, pitch angle θ, and track angle γ. Therefore, the FHV models can be decomposed into velocity subsystem (equation (1)) and altitude subsystem (equations (2)-(5)), and then the control laws are designed separately [26].


## 2.2. Theory of Fuzzy Approximation

In recent years, the universal approximation characteristics of the fuzzy system have been gradually discovered, proposed, and applied. Under sufficient rules of the fuzzy system, the fuzzy system can approximate the given nonlinear continuous function with arbitrary precision [27, 28]. The complete fuzzy system includes inference engine, knowledge base, fuzzification interface, and defuzzification interface. Compared with neural network and polynomial function approximation, the advantage of the fuzzy system is that it can effectively control based on fuzzy logic. Where the input of the system X=x1,x2,⋯,xnT∈Rn, the output is F^X.
(11)F^X∣θ=θ∗TξX.In the fuzzy system,θ∗=θ1∗,θ2∗,⋯,θp∗T∈Rp represents the parameter vector of ideal weight coefficient, fuzzy basis function ξX=ξ1X,ξ2X,⋯,ξp,XT∈Rp. Fuzzy basis functions are expressed as
(12)ξjX=∏i=1nμFijxi∑l=1N∏i=1nμFijxi.In equation (12), Fij is the jth j=1,⋯,N fuzzy rule corresponding to the ith i=1,⋯,n input point, N is the number of fuzzy rules, μFij is the membership function, 0≤μFij≤1. In this article, the Gauss basis function is chosen as the membership function.Lemma 1 [27].
Suppose that the real continuous functionFX is an arbitrary function defined on compact set Ω. For any μ>0, the fuzzy system F^X=θ∗TξX can be obtained, and the following conditions are guaranteed:
(13)supx∈ΩFX−θ∗TξX≤μ.


## 2.3. Saturation Auxiliary System

In order to maintain the normal working mode of scramjet and the physical constraint of FHV by deflection angle, the actuators of control inputsΦ and δe are limited to some extent. The saturation problem of control input is described as follows:
(14)Φc=Φmax,Φ>Φmax,Φ,Φmin≤Φ≤Φmax,Φmin,Φ<Φmin,(15)δec=δemax,δe>δemax,δe,δemin≤δe≤δemin,δe<δemin,δemax,where Φc denotes the fuel-air ratio in the case of input saturation, δec denotes the elevator deflection in the case of input saturation, Φmin and Φmax are the lower and upper bounds of Φ, and δemin and δemax are the lower and upper bounds of δe. In this article, a saturation auxiliary system is designed under the condition of input saturation. The input constraints of the control system are transformed into unconstrained ones.When the inputs of the saturated auxiliary system areΦ and δe, the sigmoid functions are used to approximate the outputs Φc and δec of the auxiliary system.
(16)Φc=Φmax−Φmin2sigmoid2Φ−Φmax+ΦminΦmax−Φmin+Φmax+Φmin2,δec=δemax−δemin2sigmoid2δe−δemin+δemaxδemax−δemin+δemax+δemin2.According to equation (16) and the characteristic of the sigmoid function, whether Φ and δe take any value, the saturation auxiliary system can make Φc and δec meet the constraints of equation (15).Remark 4.
In the controller design, the original control objective can be converted to design unconstrainedΦ and δe such that velocity Vand altitude h track their reference commands Vref and href.


## 2.4. Controller Design

In the following section, the altitude subsystem and velocity subsystem of the controller are designed with the new fuzzy-approximation-based control scheme in this article.


### 2.4.1. Controller Design of the Altitude Subsystem

In the altitude subsystem, we design an altitude controller based on nonaffine dynamics and a fuzzy approximator to achieve stable tracking of reference altitudehref.Define altitude tracking errorh~=h−href, the reference command of γ is chosen as
(17)γd=arcsin−kh,1h~−kh,2∫0th~dτ+ḣrefV,where kh,1∈R+ and kh,2∈R+ are design parameters.If the design parameterskh,1>0, kh,1>0 and γ→γd, then h~ can be regulated to zero exponentially [29]. In this way, the control objective of the altitude subsystem can be transformed to γ→γd by designing the appropriate control law δe.According to the idea expressed in [20, 21], we define a set of newly state variables; the control input δe is also viewed as the newly defined state to deal with the nonaffine features. Define new state variables
(18)ψh,1=γ,ψh,2=θ,ψh,3=Q,ψh,4=δe.Original nonaffine systems (3)-(5) can be rewritten to the following equivalent general nonaffine form according to equation (18).
(19)ψ̇h,1=Ж1ψh,1,ψh,2,ψ̇h,2=ψh,3,ψ̇h,3=Ж3ψh,1,ψh,2,ψh,3,ψh,4,ψ̇h,4=uh−τhψh,4,yh=ψh,1=γ,where Ж1ψh,1,ψh,2 and Ж3ψh,1,ψh,2,ψh,3,ψh,4 are continuous differentiable nonlinear functions; uh is an introduced variable, and it is regarded as the virtual control input of (19); the control inputs δe are generated by a low-pass filter “ψ̇h,4=uh−τhψh,4” driven by uh. “ψ̇h,4=uh−τhψh,4” is input-to-state stable. yh is the output of (19); τh∈R+is the design parameter.From equations (3)-(5) together with previous studies [17], we have
(20)∂Ж1ψh,1,ψh,2∂ψh,1≈q¯SCLα+Tcosθ−γmV>0,∂Ж3ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4≈q¯Sc¯ce+ς~2N2δe/k2Iyy>0.Next, transform the nonaffine models.Defineυ1=ψh,1=γ, υ2=υ̇1=Ж1ψh,1,ψh,2, combined with equation (19), the time derivative of υ2⁠ is derived as
(21)υ̇2=∂Ж1ψh,1,ψh,2∂ψh,1ψ̇h,1+∂Ж1ψh,1,ψh,2∂ψh,2ψ̇h,2=∂Ж1ψh,1,ψh,2∂ψh,1Ж1ψh,1,ψh,2+∂Ж1ψh,1,ψh,2∂ψh,2ψh,3≜ϒ1ψh,1,ψh,2,ψh,3.It can be noted that(22)∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3=∂Ж1ψh,1,ψh,2∂ψh,2>0.We further defineυ3=υ̇2=Y1ψh,1,ψh,2,ψh,3. Taking time derivative of υ3 and substituting (19)
(23)υ̇3=∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,1ψ̇h,1+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,2ψ̇h,2+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3ψ̇h,3=∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,1Ж1ψh,1,ψh,2+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,2ψh,3+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3Ж3ψh,1,ψh,2,ψh,3,ψh,4≜ϒ2ψh,1,ψh,2,ψh,3,ψh,4.Finally, defineυ4=υ̇3=Y2ψh,1,ψh,2,ψh,3,ψh,4. Invoking (19), the time derivative of υ4 is derived as
(24)υ̇4=∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,1ψ̇h,1+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,2ψ̇h,2+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,3ψ̇h,3+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4ψ̇h,4=∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,1Ж1ψh,1,ψh,2+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,2ψh,3+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,3Ж3ψh,1,ψh,2,ψh,3,ψh,4−∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4τhψh,4+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4uh≜ϒ3ψh,1,ψh,2,ψh,3,ψh,4+Khuh.where
(25)Κh=∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4=∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3∂Ж3ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4>0.Through model transformation, the nonaffine system (19) can be converted into affine form.
(26)υ̇1=υ2,υ̇2=υ3,υ̇3=υ4,υ̇4=Y3ψh,1,ψh,2,ψh,3,ψh,4+Κhuh,yh=υ1=γ.The fuzzy-approximation-based altitude subsystem controller is designed by using the transformed affine form.Remark 5.
According to the theory of input-to-state stability, when the input is bounded, the state of the system is also bounded. So we need to prove the boundedness ofuh.Theorem 1.
If the control inputsuh are bounded, there must be fixed values uhM, such that
(27)uh≤uhM.Proof.
Define the Lyapunov function(28)Lψh,4=ψh,422.
Taking time derivative along (28).
(29)L̇ψh,4=ψh,4ψ̇h,4=ψh,4uh−τhψh,4=−τhψh,42+uhψh,4≤−τhψh,42+uhM22+uh,422=−τh−12uh,422+uhM22
Ifτh>1/2, ψh,4 will ultimately converge to the compact Ωψh,4, where Ωψh,4=ψh,4∣ψh,4≤uhM2/2/τh−1/2. That is, ψh,4 is bounded and the low-pass filter “ψ̇h,4=uh−τhψh,4” is stable when uh is bounded.
Define tracking erroreh,1 as
(30)eh,1=υ1−υ1d.
Taking time derivative along (30) and employing (26), it leads to
(31)ėh,1=υ̇1−υ̇1d=υ2−υ̇1d.
Define intermediate variableυ2d as
(32)υ2d=−λh,1eh,1+υ̇1d,where λh,1∈R+ is the design parameter.
Then, define tracking erroreh,2 as
(33)eh,2=υ2−υ2d.
Applying (26), the time derivative of eh,2 is derived as
(34)ėh,2=υ̇2−υ̇2d=υ3−υ̇2d.
Define a new intermediate state variableυ3d as
(35)υ3d=−λh,2eh,2−eh,1+υ̇2d,where λh,2∈R+ is the design parameter.
Define tracking erroreh,3 as
(36)eh,3=υ3−υ3d.
Differentiating along (36) with respect to time and applying (26), we get
(37)ėh,3=υ̇3−υ̇3d=υ4−υ̇3d.
Then, the intermediate variableυ4d is designed as
(38)υ4d=−λh,3eh,3−eh,2+υ̇3d,where λh,3∈R+ is the design parameter.
Finally, define tracking erroreh,4 as
(39)eh,4=υ4−υ4d.
Taking the time derivative ofeh,4 and invoking (26) lead to
(40)ėh,4=υ̇4−υ̇4d=Y3ψh,1,ψh,2,ψh,3,ψh,4+Κhuh−υ̇4d=ΚhY3ψh,1,ψh,2,ψh,3,ψh,4−υ̇4dΚh+uh.
Define unknown functionFh as
(41)Fh=Y3ψh,1,ψh,2,ψh,3,ψh,4−υ̇4dΚh+uh=FhΨh,where Ψh=ψh,1,ψh,2,ψh,3,ψh,4T. Since the function Fh is an unknown function, the fuzzy function approximator designed in the above part is used to approximate it. The estimated value F^h of the nonlinear function Fh can be expressed as
(42)F^hΨh∣θh=θh∗TξhΨh,where Ψh=ψh,1,ψh,2,ψh,3,ψh,4T∈R+ is the input vector to the unknown function, fuzzy basis function ξhΨh=ξh,1Ψh,ξh,2Ψh,⋯,ξh,p1,ΨhT∈Rp1, θh∗=θh,1∗,θh,2∗,⋯,θh,p1∗T∈Rp1is the ideal weight coefficient parameter vector, and the unknown function FV can be expressed as
(43)FhΨh=θh∗TξhΨh+μh,where μh represents the estimated error, and μh≤μhM; μhM is the upper bound of the estimated error.
Finally, the control law is devised as(44)uh=−λh,4eh,4−eh,3−φ^12ξhTΨhξhΨh,where λh,4∈R+ is the design parameter. φ^1 denotes the estimation of φ1=θh∗2; its adaptive law is selected as
(45)φ^̇1=l12eh,1ξhTΨhξhΨh−2φ^1,where l1∈R+ is the design parameter.
Substitute equation (32) into equation (33) to obtain
(46)eh,2=υ2−−λh,1eh,1+υ̇1d=υ2+λh,1eh,1−υ̇1d=ėh,1+λh,1eh,1.
Taking time derivative along (46),
(47)ėh,2=e¨h,1+λh,1ėh,1.
By combining equations (26) and (35), equation (36) can be obtained.
(48)eh,3=υ3−−λh,2eh,2−eh,1+υ̇2d=υ3+λh,2eh,2+eh,1−υ̇2d=ėh,2+λh,2eh,2+eh,1.
Substituting equations (46) and (47) into equation (48), it yields
(49)eh,3=e¨h,1+λh,1ėh,1+λh,2ėh,1+λh,1eh,1+eh,1=e¨h,1+λh,1+λh,2ėh,1+λh,1λh,2+1eh,1.
Taking time derivative along (49), we obtain
(50)ėh,3=e⃛h,1λh,1+λh,2e¨h,1+λh,1λh,2+1ėh,1.
Substituting (26) and (38) into (39), eh,4 is formulated as
(51)eh,4=υ4−−λh,3eh,3−eh,2+υ̇3d=υ4+λh,3eh,3+eh,2−υ̇3d=ėh,3+λh,3eh,3+eh,2.
Combining equations (46), (49), and (50), we can obtain
(52)eh,4=e⃛h,1+λh,1+λh,2e¨h,1+λh,1λh,2+1ėh,1+λh,3e¨h,1+λh,1+λh,2ėh,1+λh,1λh,2+1eh,1+ėh,1+λh,1eh,1=e⃛h,1+λh,1+λh,2+λh,3e¨h,1+λh,1λh,2+λh,1λh,3+λh,2λh,3+2ėh,1+λh,1+λh,3+λh,1λh,2λh,3eh,1.
Substituting (49) and (52) into (44) leads to
(53)uh=−λh,4e⃛h,1−λh,4λh,1+λh,4λh,2+λh,4λh,3+1e¨h,1−λh,4λh,1λh,2+λh,4λh,2λh,3+λh,4λh,3λh,1+2λh,4+λh,1+λh,2ėh,1−λh,4λh,1λh,2λh,3+λh,4λh,3+λh,4λh,1+λh,1λh,2+1eh,1−φ^12ξhTΨhξhΨh.Remark 6.
Unlike the back-stepping control method proposed by Bu et al. [30], the presented altitude controller is exploited using an improved back-stepping framework, in which the virtual controller is transformed into intermediate variables and only the final actual control laws are needed to be implemented, which avoids repeated calculation and greatly reduces the computational pressure of the system.Remark 7.
Compared with the method in [16, 31, 32], each weight update requires a lot of calculation. In this article, only one online learning parameter is needed by using the norm estimation approach, which makes the online calculation of the system significantly reduced.Theorem 2.
Considering the closed-loop subsystem consisting of plant (26), with controller (53), and adaptive law (45). Then all the signals involved are semiglobally uniformly ultimately bounded.Proof.
Consider the following Lyapunov function:(54)Wh=eh,122+eh,222+eh,322+eh,422Κh+φ~122l1with φ~1=φ^1−φ1.
Applying (31), (34), (37), (40), (41), and (45), Ẇhis derived as
(55)Ẇh=eh,1ėh,1+eh,2ėh,2+eh,3ėh,3+eh,4ėh,4Κh+φ~1φ^̇1l1=eh,1υ2−υ̇1d+eh,2υ3−υ̇2d+eh,3υ4−υ̇3d+eh,4Y3ψh,1,ψh,2,ψh,3,ψh,4−υ̇4dΚh+uh+φ~1l1l12eh,1φ^1ξhTΨhξhΨh−2φ^1=eh,1υ2d+eh,2−υ̇1d+eh,2υ3d+eh,3−υ̇2d+eh,3υ4d+eh,4−υ̇3d+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+μh+uh−2φ~1φ^1l1.
Considering (32), (35), and (38), Ẇhbecomes
(56)Ẇh=eh,1−λh,1eh,1+υ̇1d+eh,2−υ̇1d+eh,2−λh,2eh,2−eh,1+υ̇2d+eh,3−υ̇2d+eh,3−λh,3eh,3−eh,2+eh,4+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+μh+uh−2φ~1φ^1l1.
Substituting (44) into (56) yields, we can get
(57)Ẇh=−λh,1eh,12−λh,2eh,22−λh,3eh,32+eh,3eh,4+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+eh,4μh+eh,4−λh,4eh,4−eh,3−φ^12ξhTΨhξhΨh−2φ~1φ^1l1=−λh,1eh,12−λh,2eh,22−λh,3eh,32−λh,4eh,42+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh−eh,4φ^12ξhTΨhξhΨh+eh,4μh−2φ~1φ^1l1.
From the fact that(58)φ~12+2φ~1φ^1−φ~1+φ12=φ~12+2φ~1φ1+φ12=φ~1+φ12≥0,
we obtain(59)2φ~1φ^1≥φ~12−φ12.
Thereby, equation (57) becomes
(60)Ẇh≤−λh,1eh,12−λh,2eh,22−λh,3eh,32−λh,4eh,42+12eh,1φ~1ξhTΨhξhΨh+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+eh,4μh−eh,4φ~12ξhTΨhξhΨh−eh,4φ12ξhTΨhξhΨh−φ~12l1+φ12l1.
In this article, since0≤μFijx≤1, and applying (12), we know that 0≤ξjX≤1. There exists a constantξ¯h∈R+, such that ξhΨh≤ξ¯h. Hence, we get
(61)12eh,1φ~1ξhTΨhξhΨh≤12ξ¯h2eh,1φ~1≤2l1212ξ¯h2eh,12+φ~1222l1=l1ξ¯h44eh,12+φ~124l1,eh,4θh∗TξhΨh≤eh,4φ1ξ¯h≤φ1ξ¯heh,422+12=φ1ξ¯h2eh,42+φ1ξ¯h2,eh,4μh≤μhMeh,4≤μhMeh,422+12=μhMeh,422+μhM2−eh,4φ~12ξhTΨhξhΨh≤12ξ¯h2eh,4φ~1≤2l1212ξ¯h2eh,42+φ~1222l1=l1ξ¯h44eh,42+φ~124l1−eh,4φ12ξhTΨhξhΨh≤12ξ¯h2eh,4φ1≤12φ1ξ¯h2eh,422+12=φ1ξ¯h24eh,42+φ1ξ¯h24.
Then, (60) becomes
(62)Ẇh≤−λh,1eh,12−λh,2eh,22−λh,3eh,32−λh,4eh,42+l1ξ¯h44eh,12+φ~124l1+φ1ξ¯h2eh,42+φ1ξ¯h2+μhMeh,422+μhM2+l1ξ¯h44eh,42+φ~124l1+φ1ξ¯h24eh,42+φ1ξ¯h24−φ~12l1+φ12l1,Ẇh≤−λh,1−l1ξ¯h44eh,12−λh,2eh,22−λh,3eh,32+φ1ξ¯h2+μhM2+φ1ξ¯h24−λh,4−φ1ξ¯h2−μhM2−l1ξ¯h44−φ1ξ¯h24eh,42+φ12l1−φ~122l1.
Letλh,1>l1ξ¯h4/4, λh,4>φ1ξ¯h/2+μhM/2+l1ξ¯h4/4+φ1ξ¯h2/4, λh,2>0, λh,3>0, and l1>0. And define the following compact sets
(63)Ωeh,1=eh,1∣eh,1≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,1−l1ξ¯h44,Ωeh,4=eh,4∣eh,4≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,4−φ1ξ¯h2−μhM2−l1ξ¯h44−φ1ξ¯h24,Ωeh,2=eh,2∣eh,2≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,2,Ωeh,3=eh,3∣eh,3≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,3,Ωφ~1=φ~1∣φ~1≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/12l1.
Ifeh,1∉Ωeh,1 or eh,4∉Ωeh,4 or eh,2∉Ωeh,2 or eh,3∉Ωeh,3 or φ~1∉Ωφ~1, Ẇh will be negative. It is clear that eh,1, eh,4, eh,2, eh,3 and φ~1 are semiglobally uniformly ultimately bounded. This completes the proof.


## 2.4.1. Controller Design of the Altitude Subsystem

In the altitude subsystem, we design an altitude controller based on nonaffine dynamics and a fuzzy approximator to achieve stable tracking of reference altitudehref.Define altitude tracking errorh~=h−href, the reference command of γ is chosen as
(17)γd=arcsin−kh,1h~−kh,2∫0th~dτ+ḣrefV,where kh,1∈R+ and kh,2∈R+ are design parameters.If the design parameterskh,1>0, kh,1>0 and γ→γd, then h~ can be regulated to zero exponentially [29]. In this way, the control objective of the altitude subsystem can be transformed to γ→γd by designing the appropriate control law δe.According to the idea expressed in [20, 21], we define a set of newly state variables; the control input δe is also viewed as the newly defined state to deal with the nonaffine features. Define new state variables
(18)ψh,1=γ,ψh,2=θ,ψh,3=Q,ψh,4=δe.Original nonaffine systems (3)-(5) can be rewritten to the following equivalent general nonaffine form according to equation (18).
(19)ψ̇h,1=Ж1ψh,1,ψh,2,ψ̇h,2=ψh,3,ψ̇h,3=Ж3ψh,1,ψh,2,ψh,3,ψh,4,ψ̇h,4=uh−τhψh,4,yh=ψh,1=γ,where Ж1ψh,1,ψh,2 and Ж3ψh,1,ψh,2,ψh,3,ψh,4 are continuous differentiable nonlinear functions; uh is an introduced variable, and it is regarded as the virtual control input of (19); the control inputs δe are generated by a low-pass filter “ψ̇h,4=uh−τhψh,4” driven by uh. “ψ̇h,4=uh−τhψh,4” is input-to-state stable. yh is the output of (19); τh∈R+is the design parameter.From equations (3)-(5) together with previous studies [17], we have
(20)∂Ж1ψh,1,ψh,2∂ψh,1≈q¯SCLα+Tcosθ−γmV>0,∂Ж3ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4≈q¯Sc¯ce+ς~2N2δe/k2Iyy>0.Next, transform the nonaffine models.Defineυ1=ψh,1=γ, υ2=υ̇1=Ж1ψh,1,ψh,2, combined with equation (19), the time derivative of υ2⁠ is derived as
(21)υ̇2=∂Ж1ψh,1,ψh,2∂ψh,1ψ̇h,1+∂Ж1ψh,1,ψh,2∂ψh,2ψ̇h,2=∂Ж1ψh,1,ψh,2∂ψh,1Ж1ψh,1,ψh,2+∂Ж1ψh,1,ψh,2∂ψh,2ψh,3≜ϒ1ψh,1,ψh,2,ψh,3.It can be noted that(22)∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3=∂Ж1ψh,1,ψh,2∂ψh,2>0.We further defineυ3=υ̇2=Y1ψh,1,ψh,2,ψh,3. Taking time derivative of υ3 and substituting (19)
(23)υ̇3=∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,1ψ̇h,1+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,2ψ̇h,2+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3ψ̇h,3=∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,1Ж1ψh,1,ψh,2+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,2ψh,3+∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3Ж3ψh,1,ψh,2,ψh,3,ψh,4≜ϒ2ψh,1,ψh,2,ψh,3,ψh,4.Finally, defineυ4=υ̇3=Y2ψh,1,ψh,2,ψh,3,ψh,4. Invoking (19), the time derivative of υ4 is derived as
(24)υ̇4=∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,1ψ̇h,1+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,2ψ̇h,2+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,3ψ̇h,3+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4ψ̇h,4=∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,1Ж1ψh,1,ψh,2+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,2ψh,3+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,3Ж3ψh,1,ψh,2,ψh,3,ψh,4−∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4τhψh,4+∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4uh≜ϒ3ψh,1,ψh,2,ψh,3,ψh,4+Khuh.where
(25)Κh=∂ϒ2ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4=∂ϒ1ψh,1,ψh,2,ψh,3∂ψh,3∂Ж3ψh,1,ψh,2,ψh,3,ψh,4∂ψh,4>0.Through model transformation, the nonaffine system (19) can be converted into affine form.
(26)υ̇1=υ2,υ̇2=υ3,υ̇3=υ4,υ̇4=Y3ψh,1,ψh,2,ψh,3,ψh,4+Κhuh,yh=υ1=γ.The fuzzy-approximation-based altitude subsystem controller is designed by using the transformed affine form.Remark 5.
According to the theory of input-to-state stability, when the input is bounded, the state of the system is also bounded. So we need to prove the boundedness ofuh.Theorem 1.
If the control inputsuh are bounded, there must be fixed values uhM, such that
(27)uh≤uhM.Proof.
Define the Lyapunov function(28)Lψh,4=ψh,422.
Taking time derivative along (28).
(29)L̇ψh,4=ψh,4ψ̇h,4=ψh,4uh−τhψh,4=−τhψh,42+uhψh,4≤−τhψh,42+uhM22+uh,422=−τh−12uh,422+uhM22
Ifτh>1/2, ψh,4 will ultimately converge to the compact Ωψh,4, where Ωψh,4=ψh,4∣ψh,4≤uhM2/2/τh−1/2. That is, ψh,4 is bounded and the low-pass filter “ψ̇h,4=uh−τhψh,4” is stable when uh is bounded.
Define tracking erroreh,1 as
(30)eh,1=υ1−υ1d.
Taking time derivative along (30) and employing (26), it leads to
(31)ėh,1=υ̇1−υ̇1d=υ2−υ̇1d.
Define intermediate variableυ2d as
(32)υ2d=−λh,1eh,1+υ̇1d,where λh,1∈R+ is the design parameter.
Then, define tracking erroreh,2 as
(33)eh,2=υ2−υ2d.
Applying (26), the time derivative of eh,2 is derived as
(34)ėh,2=υ̇2−υ̇2d=υ3−υ̇2d.
Define a new intermediate state variableυ3d as
(35)υ3d=−λh,2eh,2−eh,1+υ̇2d,where λh,2∈R+ is the design parameter.
Define tracking erroreh,3 as
(36)eh,3=υ3−υ3d.
Differentiating along (36) with respect to time and applying (26), we get
(37)ėh,3=υ̇3−υ̇3d=υ4−υ̇3d.
Then, the intermediate variableυ4d is designed as
(38)υ4d=−λh,3eh,3−eh,2+υ̇3d,where λh,3∈R+ is the design parameter.
Finally, define tracking erroreh,4 as
(39)eh,4=υ4−υ4d.
Taking the time derivative ofeh,4 and invoking (26) lead to
(40)ėh,4=υ̇4−υ̇4d=Y3ψh,1,ψh,2,ψh,3,ψh,4+Κhuh−υ̇4d=ΚhY3ψh,1,ψh,2,ψh,3,ψh,4−υ̇4dΚh+uh.
Define unknown functionFh as
(41)Fh=Y3ψh,1,ψh,2,ψh,3,ψh,4−υ̇4dΚh+uh=FhΨh,where Ψh=ψh,1,ψh,2,ψh,3,ψh,4T. Since the function Fh is an unknown function, the fuzzy function approximator designed in the above part is used to approximate it. The estimated value F^h of the nonlinear function Fh can be expressed as
(42)F^hΨh∣θh=θh∗TξhΨh,where Ψh=ψh,1,ψh,2,ψh,3,ψh,4T∈R+ is the input vector to the unknown function, fuzzy basis function ξhΨh=ξh,1Ψh,ξh,2Ψh,⋯,ξh,p1,ΨhT∈Rp1, θh∗=θh,1∗,θh,2∗,⋯,θh,p1∗T∈Rp1is the ideal weight coefficient parameter vector, and the unknown function FV can be expressed as
(43)FhΨh=θh∗TξhΨh+μh,where μh represents the estimated error, and μh≤μhM; μhM is the upper bound of the estimated error.
Finally, the control law is devised as(44)uh=−λh,4eh,4−eh,3−φ^12ξhTΨhξhΨh,where λh,4∈R+ is the design parameter. φ^1 denotes the estimation of φ1=θh∗2; its adaptive law is selected as
(45)φ^̇1=l12eh,1ξhTΨhξhΨh−2φ^1,where l1∈R+ is the design parameter.
Substitute equation (32) into equation (33) to obtain
(46)eh,2=υ2−−λh,1eh,1+υ̇1d=υ2+λh,1eh,1−υ̇1d=ėh,1+λh,1eh,1.
Taking time derivative along (46),
(47)ėh,2=e¨h,1+λh,1ėh,1.
By combining equations (26) and (35), equation (36) can be obtained.
(48)eh,3=υ3−−λh,2eh,2−eh,1+υ̇2d=υ3+λh,2eh,2+eh,1−υ̇2d=ėh,2+λh,2eh,2+eh,1.
Substituting equations (46) and (47) into equation (48), it yields
(49)eh,3=e¨h,1+λh,1ėh,1+λh,2ėh,1+λh,1eh,1+eh,1=e¨h,1+λh,1+λh,2ėh,1+λh,1λh,2+1eh,1.
Taking time derivative along (49), we obtain
(50)ėh,3=e⃛h,1λh,1+λh,2e¨h,1+λh,1λh,2+1ėh,1.
Substituting (26) and (38) into (39), eh,4 is formulated as
(51)eh,4=υ4−−λh,3eh,3−eh,2+υ̇3d=υ4+λh,3eh,3+eh,2−υ̇3d=ėh,3+λh,3eh,3+eh,2.
Combining equations (46), (49), and (50), we can obtain
(52)eh,4=e⃛h,1+λh,1+λh,2e¨h,1+λh,1λh,2+1ėh,1+λh,3e¨h,1+λh,1+λh,2ėh,1+λh,1λh,2+1eh,1+ėh,1+λh,1eh,1=e⃛h,1+λh,1+λh,2+λh,3e¨h,1+λh,1λh,2+λh,1λh,3+λh,2λh,3+2ėh,1+λh,1+λh,3+λh,1λh,2λh,3eh,1.
Substituting (49) and (52) into (44) leads to
(53)uh=−λh,4e⃛h,1−λh,4λh,1+λh,4λh,2+λh,4λh,3+1e¨h,1−λh,4λh,1λh,2+λh,4λh,2λh,3+λh,4λh,3λh,1+2λh,4+λh,1+λh,2ėh,1−λh,4λh,1λh,2λh,3+λh,4λh,3+λh,4λh,1+λh,1λh,2+1eh,1−φ^12ξhTΨhξhΨh.Remark 6.
Unlike the back-stepping control method proposed by Bu et al. [30], the presented altitude controller is exploited using an improved back-stepping framework, in which the virtual controller is transformed into intermediate variables and only the final actual control laws are needed to be implemented, which avoids repeated calculation and greatly reduces the computational pressure of the system.Remark 7.
Compared with the method in [16, 31, 32], each weight update requires a lot of calculation. In this article, only one online learning parameter is needed by using the norm estimation approach, which makes the online calculation of the system significantly reduced.Theorem 2.
Considering the closed-loop subsystem consisting of plant (26), with controller (53), and adaptive law (45). Then all the signals involved are semiglobally uniformly ultimately bounded.Proof.
Consider the following Lyapunov function:(54)Wh=eh,122+eh,222+eh,322+eh,422Κh+φ~122l1with φ~1=φ^1−φ1.
Applying (31), (34), (37), (40), (41), and (45), Ẇhis derived as
(55)Ẇh=eh,1ėh,1+eh,2ėh,2+eh,3ėh,3+eh,4ėh,4Κh+φ~1φ^̇1l1=eh,1υ2−υ̇1d+eh,2υ3−υ̇2d+eh,3υ4−υ̇3d+eh,4Y3ψh,1,ψh,2,ψh,3,ψh,4−υ̇4dΚh+uh+φ~1l1l12eh,1φ^1ξhTΨhξhΨh−2φ^1=eh,1υ2d+eh,2−υ̇1d+eh,2υ3d+eh,3−υ̇2d+eh,3υ4d+eh,4−υ̇3d+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+μh+uh−2φ~1φ^1l1.
Considering (32), (35), and (38), Ẇhbecomes
(56)Ẇh=eh,1−λh,1eh,1+υ̇1d+eh,2−υ̇1d+eh,2−λh,2eh,2−eh,1+υ̇2d+eh,3−υ̇2d+eh,3−λh,3eh,3−eh,2+eh,4+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+μh+uh−2φ~1φ^1l1.
Substituting (44) into (56) yields, we can get
(57)Ẇh=−λh,1eh,12−λh,2eh,22−λh,3eh,32+eh,3eh,4+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+eh,4μh+eh,4−λh,4eh,4−eh,3−φ^12ξhTΨhξhΨh−2φ~1φ^1l1=−λh,1eh,12−λh,2eh,22−λh,3eh,32−λh,4eh,42+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh−eh,4φ^12ξhTΨhξhΨh+eh,4μh−2φ~1φ^1l1.
From the fact that(58)φ~12+2φ~1φ^1−φ~1+φ12=φ~12+2φ~1φ1+φ12=φ~1+φ12≥0,
we obtain(59)2φ~1φ^1≥φ~12−φ12.
Thereby, equation (57) becomes
(60)Ẇh≤−λh,1eh,12−λh,2eh,22−λh,3eh,32−λh,4eh,42+12eh,1φ~1ξhTΨhξhΨh+12eh,1φ~1ξhTΨhξhΨh+eh,4θh∗TξhΨh+eh,4μh−eh,4φ~12ξhTΨhξhΨh−eh,4φ12ξhTΨhξhΨh−φ~12l1+φ12l1.
In this article, since0≤μFijx≤1, and applying (12), we know that 0≤ξjX≤1. There exists a constantξ¯h∈R+, such that ξhΨh≤ξ¯h. Hence, we get
(61)12eh,1φ~1ξhTΨhξhΨh≤12ξ¯h2eh,1φ~1≤2l1212ξ¯h2eh,12+φ~1222l1=l1ξ¯h44eh,12+φ~124l1,eh,4θh∗TξhΨh≤eh,4φ1ξ¯h≤φ1ξ¯heh,422+12=φ1ξ¯h2eh,42+φ1ξ¯h2,eh,4μh≤μhMeh,4≤μhMeh,422+12=μhMeh,422+μhM2−eh,4φ~12ξhTΨhξhΨh≤12ξ¯h2eh,4φ~1≤2l1212ξ¯h2eh,42+φ~1222l1=l1ξ¯h44eh,42+φ~124l1−eh,4φ12ξhTΨhξhΨh≤12ξ¯h2eh,4φ1≤12φ1ξ¯h2eh,422+12=φ1ξ¯h24eh,42+φ1ξ¯h24.
Then, (60) becomes
(62)Ẇh≤−λh,1eh,12−λh,2eh,22−λh,3eh,32−λh,4eh,42+l1ξ¯h44eh,12+φ~124l1+φ1ξ¯h2eh,42+φ1ξ¯h2+μhMeh,422+μhM2+l1ξ¯h44eh,42+φ~124l1+φ1ξ¯h24eh,42+φ1ξ¯h24−φ~12l1+φ12l1,Ẇh≤−λh,1−l1ξ¯h44eh,12−λh,2eh,22−λh,3eh,32+φ1ξ¯h2+μhM2+φ1ξ¯h24−λh,4−φ1ξ¯h2−μhM2−l1ξ¯h44−φ1ξ¯h24eh,42+φ12l1−φ~122l1.
Letλh,1>l1ξ¯h4/4, λh,4>φ1ξ¯h/2+μhM/2+l1ξ¯h4/4+φ1ξ¯h2/4, λh,2>0, λh,3>0, and l1>0. And define the following compact sets
(63)Ωeh,1=eh,1∣eh,1≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,1−l1ξ¯h44,Ωeh,4=eh,4∣eh,4≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,4−φ1ξ¯h2−μhM2−l1ξ¯h44−φ1ξ¯h24,Ωeh,2=eh,2∣eh,2≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,2,Ωeh,3=eh,3∣eh,3≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/λh,3,Ωφ~1=φ~1∣φ~1≤φ1ξ¯h2+μhM2+φ12l1+φ1ξ¯h24/12l1.
Ifeh,1∉Ωeh,1 or eh,4∉Ωeh,4 or eh,2∉Ωeh,2 or eh,3∉Ωeh,3 or φ~1∉Ωφ~1, Ẇh will be negative. It is clear that eh,1, eh,4, eh,2, eh,3 and φ~1 are semiglobally uniformly ultimately bounded. This completes the proof.


## 2.5. Controller Design of Velocity Subsystem

For the velocity subsystem, its tracking target is to achieve the stable tracking of velocityV to reference velocity Vref.Define new state variablesψV,1=VandψV,2=Φ. In order to ensure the control effect, the velocity subsystem should be changed into an equivalent general nonaffine form with ψV,1 and ψV,2.
(64)ψ̇V,1=ΓψV,1,ψV,2,ψ̇V,2=uV−τVψV,2,yV=ψV,1,where the transformation consists of a low-pass filter “ψ̇V,2=uV−τVψV,2” and 2 newly defined state variables. ΓψV,1,ψV,2 is an unknown continuous differentiable nonlinear function, uV is the input of (64), and yV is the output of (64); τV∈R+ is the design parameter.Then, define state variablesε1 and ε2. Where ε1=yV, ε2=ε̇1=ψ̇V,1=ΓψV,1,ψV,2, we can get
(65)ε̇2=∂ΓψV,1,ψV,2∂ψV,1ψ̇V,1+∂ΓψV,1,ψV,2∂ψV,2ψ̇V,2=∂ΓψV,1,ψV,2∂ψV,1ΓψV,1,ψV,2+∂ΓψV,1,ψV,2∂ψV,2uV−τVψV,2=∂ΓψV,1,ψV,2∂ψV,1ΓψV,1,ψV,2+∂ΓψV,1,ψV,2∂ψV,2uV−τV∂ΓψV,1,ψV,2∂ψV,2ψV,2≜YVψV,1,ψV,2=∂ΓψV,1,ψV,2∂ψV,1ΓψV,1,ψV,2+ΚVuV,Where nonlinear continuous functions YVψV,1,ψV,2 and КV are expressed as
(66)YVψV,1,ψV,2=∂ΓψV,1,ψV,2∂ψV,1ΓψV,1,ψV,2τV∂ΓψV,1,ψV,2∂ψV,2ψV,2,ΚV=∂ΓψV,1,ψV,2∂ψV,2=∂Tcosα−D/m−ginγ∂Φ=cosαmβ1h,q¯α3+β3h,q¯α2+β5h,q¯α+β7h,q¯.Remark 8.
Noting thatα∈−8.7rad,8.7rad,×10−2 α∈−8.7rad,8.7rad×10−2, so we can get the definitions of α2 and α3 as α2∈0rad2,7.57rad2×10−3 and α3∈−6.5850rad3,6.5850rad3×10−4, respectively. And we can calculate that КV>0.According to the above analysis, we can express the nonaffine model in the form of the affine model, as shown in equation (67).
(67)ε̇1=ε2,ε̇2=YVψV,1,ψV,2+ΚVuV,yV=ψV,1.Next, we need to prove the boundedness ofuV.Theorem 3.
If the control inputuV is bounded, there must be a fixed value uVM, such that
(68)uV≤uVM.Proof.
Construct the Lyapunov function(69)LψV,2=ψV,222.
Taking time derivative along (69)
(70)L̇ψV,2=ψV,2ψ̇V,2=ψV,2uV−τVψV,2=−τVψV,22+uVψV,2≤−τVψV,22+uVM22+ψV,222=−τV−12ψV,222+uVM22.
IfτV>1/2,ψV,2 will ultimately converge to the compact ΩψV,2. Where ΩψV,2=ψV,2∣ψV,2≤uVM2/2/τV−1/2. As a result, the boundedness of uh leads to the newly introduced system “ψ̇V,2=uV−τVψV,2” which is stable.
Define tracking erroreV,1 as
(71)eV,1=ε1−ε1d=ε1−Vref.
Taking time derivative along (71) and employing (67), it leads to
(72)ėV,1=ε̇1−ε̇1d=ε2−ε̇1d.
Define intermediate variableε2d as
(73)ε2d=−λV,1eV,1+ε̇1d,where λV,1∈R+ is the design parameter.
Define tracking erroreV,2 as
(74)eV,2=ε2−ε2d.
Utilizing (67) and (73), the time derivative of eV,2 is derived as
(75)ėV,2=ε̇2−ε̇2d=YVψV,1,ψV,2+ΚVuV−ε̇2d=ΚVYVψV,1,ψV,2−ε̇2dΚV+uV≜ΚVFVΨV,where ΨV=ψV,1,ψV,2T. Since the function FV is an unknown function, the fuzzy function approximator is used to approximate it. The estimated value F^V of the nonlinear function FV can be expressed as
(76)F^VΨV∣θV=θV∗TξVΨV.
ΨV=ψV,1,ψV,2T∈R+ is the input vector of the unknown function; θV∗=θV,1∗,θV,2∗,⋯,θV,p2∗T∈Rp2 is the parameter vector of ideal weight coefficient, fuzzy basis function ξVΨV=ξV,1ΨV,ξV,2ΨV,⋯,ξV,p2,ΨVT∈Rp2. The unknown function FV can be expressed as
(77)FVΨV=θV∗TξVΨV+μV,where μV stands for estimation error, μV≤μVM; μVM represents the upper bound of estimation error.
The velocity control law is designed as(78)uV=−λV,2eV,2−eV,1−φ^22ξVTΨVξVΨV,where φ^2=θV∗2, and its adaptive law is chosen to be
(79)φ^̇2=l22eV,1ξVTΨVξVΨV−2φ^2,where l2∈R+ is the design parameter.
Substituting equations (67) and (73) into equation (74), we can obtain
(80)eV,2=ε2−ε2d=ε̇1−ε2d=ε̇1−−λV,1eV,1+ε̇1d=ε̇1−ε̇1d+λV,1eV,1=eV,1+λV,1eV,1.
Substitute equation (80) into equation (78) to get
(81)uV=−λV,2eV,1+λV,1eV,1−eV,1−φ^22ξVTΨVξVΨV=−λV,2ėV,1−λV,1λV,2eV,1−φ^22ξVTΨVξVΨV.Theorem 4.
Considering the closed-loop subsystem consisting of plant (67), with controller (81) and adaptive law (79). Then, all the signals involved are semiglobally uniformly ultimately bounded.Proof.
Consider the following Lyapunov function candidate:(82)WV=eV,122+eV,222ΚV+φ~222l2with φ~2=φ^2−φ2.
Applying (73), (74) into (72), ẆV is expressed as
(83)ẆV=eV,1ėV,1+eV,2ėV,2ΚV+φ~2φ^̇2l2=eV,1eV,2−λV,1eV,1+eV,2ėV,2ΚV+φ~2φ^̇2l2.
Considering (75), (77), and (79), ẆV becomes
(84)ẆV=eV,1eV,2−λV,1eV,1+eV,2YVψV,1,ψV,2−ε̇2dΚV+uV+φ~2φ^̇2l2=eV,1eV,2−λV,1eV,12+eV,2θV∗TξVΨV+μV+uV+12eV,1φ~2ξVTΨVξVΨV−2φ~2φ^2l2.
To simplify equation (84) and substituting (78) into (84) yields, we can get
(85)ẆV=eV,1eV,2−λV,1eV,12+eV,2θV∗TξVΨV+μV−λV,2eV,2−eV,1−φ^22ξVTΨVξVΨV+12eV,1φ~2ξVTΨVξVΨV−2φ~2φ^2l2=−λV,1eV,12−λV,2eV,22+12eV,1φ~2ξVTΨVξVΨV+eV,2θV∗TξVΨV+eV,2μV−eV,2φ~22ξVTΨVξVΨV−eV,2φ22ξVTΨVξVΨV−2φ~2φ^2l2.
From the fact that(86)φ~22+2φ~2φ^2−φ~2+φ22=φ~22+2φ~2φ2+φ22=φ~2+φ22≥0,
we obtain(87)2φ~2φ^2≥φ~22−φ22.
Thereby, equation (85) becomes
(88)ẆV≤−λV,1eV,12−λV,2eV,22+12eV,1φ~2ξVTΨVξVΨV+eV,2θV∗TξVΨV−eV,2φ~22ξVTΨVξVΨV−eV,2φ22ξVTΨVξVΨV+eV,2μV−φ~22l2+φ22l2.
In this article, since0≤μFijx≤1, and applying (12), we know that 0≤ξjX≤1.There exists a constantξ¯V∈R+, such that ξVΨV≤ξ¯V. Hence, we get
(89)12eV,1φ~2ξVTΨVξVΨV≤12ξ¯V2eV,1φ~2≤2l2212ξ¯V2eV,12+φ~2222l2=l2ξ¯V44eV,12+φ~224l2,eV,2θV∗TξVΨV≤eV,2φ2ξ¯V≤φ2ξ¯VeV,222+12=φ2ξ¯V2eV,22+φ2ξ¯V2,eV,2μV≤μhMeV,2≤μVMeV,222+12=μVMeV,222+μVM2−eV,2φ~22ξVTΨVξVΨV≤12ξ¯V2eV,2φ~2≤2l2212ξ¯V2eV,22+φ~2222l2=l2ξ¯V44eV,22+φ~224l2−eV,2φ22ξVTΨVξVΨV≤12ξ¯V2eV,2φ2≤12φ2ξ¯V2eV,222+12=φ2ξ¯V24eV,22+φ2ξ¯V24.
Then, (88) becomes
(90)ẆV≤−λV,1eV,12−λV,2eV,22+l2ξ¯V44eV,12+φ~224l2+φ2ξ¯V2eV,22+φ2ξ¯V2+μVMeV,222+μVM2+l2ξ¯V44eV,22+φ~224l2+φ2ξ¯V24eV,22+φ2ξ¯V24−φ~22l2+φ22l2,ẆV≤−λV,1−l2ξ¯V44eV,12−λV,2−φ2ξ¯V2−μVM2−l2ξ¯V44−φ2ξ¯V24eV,22+φ2ξ¯V2+μVM2+φ2ξ¯V24+φ22l2−φ~222l2.
LetλV,1>l2ξ¯V4/4, λV,2>φ2ξ¯V/2+μVM/2+l2ξ¯V4/4+φ2ξ¯V2/4, l2>0. And define the following compact sets:
(91)ΩeV,1=eV,1∣eV,1≤φ2ξ¯V2+μVM2+φ22l2+φ2ξ¯V24/λV,1−l2ξ¯V44,ΩeV,2=eV,2∣eV,2≤φ2ξ¯V2+μVM2+φ22l2+φ2ξ¯V24/λV,2−φ2ξ¯V2−μVM2−l2ξ¯V44−φ2ξ¯V24,Ωφ~2=φ~2∣φ~2≤φ2ξ¯V2+μVM2+φ22l2+φ2ξ¯V24/12l2.
IfeV,1∉ΩeV,1 or eV,2∉ΩeV,2 or φ~2∉Ωφ~2, ẆV will be negative. Thus, eV,1, eV,2, and φ~2 are semiglobally uniformly ultimately bounded. This is the end of the proof.


## 3. Simulation Results

To verify the effectiveness of the new control strategy proposed in this article, the proposed controller is tested in the simulation adopting the nonlinear models (1)-(7) implemented in MATLAB/SIMULINKs. The simulation step size is fixed, the step size is 0.01 s, and the simulation period is 100 s. The initial trim conditions of the FHV are shown in Table 1. The velocity step ΔV=100m/s, altitude step Δh=200. Both the velocity and altitude reference inputs are given by a second-order reference model with a damping ratio of 0.95 and a natural frequency of 0.03rad/s. All the aerodynamic coefficients and parameters are the same as [17, 23].Table 1
Initial trim conditions.
Item
Value
Units
V
2500
m/s
h
27000
m
γ
0
deg
θ
1.5295
deg
Q
0
deg/s
η1
0.2857
—
η2
0.2857
—Design parameters are chosen askh,1=2, kh,2=0.8, τh=75, λh,1=45, λh,2=45, λh,3=53, λh,4=53, l1=0.05, τV=15, λV,1=10, λV,2=2, and l2=0.05. In order to verify the robustness of the system, it is assumed that the aerodynamic coefficients of the FHV models have a perturbation of ±40%.
(92)C=C0,0s≤t<50s,C01+0.4sin0.1πt,50s≤t<100s,where C is the value of uncertain coefficient; C0⁠ denotes the nominal value of C.The fuzzy function approximators choose the Gauss basis functions as the membership functions, set the fuzzy set of each variable to 100. The input vectors areΨh=ψh,1,ψh,2,ψh,3,ψh,4T, ΨV=ψV,1,ψV,2T. Where the fuzzy centers iV, iγ, iθ, iQ, iΦ and iδe of each fuzzy set are uniformly distributed on 2500m,2800m, −1.1°,+1.1°, 0°,11.5°, −5.7°/s,+5.7°/s, 0,2, and −23°,+23°, respectively. The membership functions of the above variables are shown in Table 2. To show the superiority of the proposed method, we do the simulation study compared with recent research methods of others. The following two different cases are considered.Case 1.
The method proposed in this paper (controller 1) is compared with the novel back-stepping control method (controller 2) in [33] without adding the saturation auxiliary system.Table 2
Membership functions of variables.
V
γ
θ
Q
Φ
δe
exp−V−iV2002
exp−γ−iγ0.052
exp−θ−iθ0.092
exp−Q−iQ0.052
exp−Φ−iΦ102
exp−δe−iδe0.052The obtained simulation results are shown in Figures1–7. As can be seen from Figures 1 and 2, both of the two methods can achieve the desired tracking effects, but the tracking accuracy of controller 1 is higher than that of controller 2 in both altitude tracking effects and velocity tracking effects. According to Figure 3, the attitude angle changes of the two methods are roughly the same, but the change amplitude of controller 1 is less than that of controller 2. Figure 4 shows that the trend of Φ and δe are similar, the lines are relatively smooth, there is no high-frequency chattering phenomenon, and the amplitude of controller 1 is small. Figure 5 shows that uV and uh⁠ are bounded. Finally, variation curves of flexible states and the estimations of φ1 and φ2 are presented in Figures 6 and 7.Case 2.
Assuming that the actuators are subject to theoretical constraints:Φ∈0.05,1.2, δe∈−18°,18°. The closed-loop system simulation of FHV is carried out by using the method presented in this paper (controller 1) and the method of adaptive neural control with nonaffine dynamics (controller 2) in [34].Figure 1
Altitude tracking performance.Figure 2
Velocity tracking performance.Figure 3
Responses of attitude angles.Figure 4
Φ and δe.Figure 5
uV and uh.Figure 6
Variation curves of flexible states.Figure 7
Estimations ofφ1 and φ2.When the actuators are constrained, the simulation results using the two methods are shown in Figures8–14. From Figures 8 and 9, it can be seen that both two methods can achieve stable tracking of reference altitude and velocity, but the altitude tracking error and velocity tracking error in controller 2 are relatively large, the maximum altitude tracking error is more than 4m, and the maximum velocity tracking error is close to 2m/s. At the same time, it can be seen from Figure 10 that the attitude angle changes of controller 1 are obviously smoother than that of controller 2. In Figure 11, although controller 2 can maintain the robustness of the system, the actuators are saturated for a long time. When controller 1 is saturated, it can also rely on the auxiliary system to maintain the stability of the system. In Figure 12, uV and uh⁠ are bounded. Variation curves of flexible states and the estimations of φ1 and φ2 are presented in Figures 13 and 14. To sum up, in both cases, the superiority of the proposed method can be demonstrated.Figure 8
Altitude tracking performance.Figure 9
Velocity tracking performance.Figure 10
Responses of attitude angles.Figure 11
Φ and δe.Figure 12
uV and uh.Figure 13
Variation curves of flexible states.Figure 14
Estimations ofφ1 and φ2.


## 4. Conclusions

(1)
In this paper, a fuzzy-approximation-based novel back-stepping control method is proposed. The method does not need virtual control laws, but only needs a final actual control law, and the control algorithm is more simplified
(2)
For each subsystem of FHV, only a fuzzy approximator is needed to approximate the total uncertainties in the subsystem. Meanwhile, the norm estimation approach is introduced to directly estimate the norm of the weight vector in the fuzzy approximator, instead of the element itself, which reduces the computational complexity of the control algorithm and guarantees the real-time performance of the algorithm
(3)
The control method in this paper is based on nonaffine models and uses low-pass filters to deal with nonaffine dynamics, avoiding the loss of dynamic characteristics in affine models
(4)
The saturation problem of actuators in nonaffine models is discussed. By designing an auxiliary system, the original control problem with input constraints is transformed into a new control problem without input constraints
(5)
In two cases, the simulation results are compared with the recent methods to verify the advantages of the proposed method



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*Source: 2607195-2019-06-24.xml*