# Backstepping Control Based on Constrained Command Filter for Hypersonic Flight Vehicles with AOA and Actuator Constraints

Authors: Junbao Wei; Haiyan Li; Ming Guo; Jing Li; Huang Huang
Journal: International Journal of Aerospace Engineering
 (2021)

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## Abstract

An antisaturation backstepping control scheme based on constrained command filter for hypersonic flight vehicle (HFV) is proposed with the consideration of angle of attack (AOA) constraint and actuator constraints of amplitude and rate. Firstly, the HFV system model is divided into velocity subsystem and height subsystem. Secondly, to handle AOA constraint, a constrained command filter is constructed to limit the amplitude of the AOA command and retain its differentiability. And the constraint range is set in advance via a prescribed performance method to guarantee that the tracking error of the AOA meets the constraint conditions and transient and steady performance. Thirdly, the proposed constrained command filter is combined with the auxiliary system for actuator constraints, which ensures that the control input meets the limited requirements of amplitude and rate, and the system is stable. In addition, the tracking errors of the system are proved to be ultimately uniformly bounded based on the Lyapunov stability theory. Finally, the effectiveness of the proposed method is verified by simulation.


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## Body

## 1. Introduction

Hypersonic flight vehicle (HFV) has attracted wide attention in the world due to its great application prospect in military and civil fields. As the key technology to develop HFV, flight control technology can realize tracking trajectory and stable flight. Therefore, the research on the controller design of HFV is significant to its development.HFV is characterized by large flight envelope, complex flight characteristics, and variable external environment and is strong coupled, nonlinear, and uncertain [1]. It is challenging to design a controller for HFV. For the problem of nonminimum phase, Ref. [2] proposes a control-oriented model which ignored the lift-lift coupling. At present, the commonly used idea is to design controllers for the input-output subsystem based on output redefinition method [3, 4]. An alternative method which is available for dealing with the controller design problem of such a strong nonlinear system as HFV is backstepping approach [5–11]. The backstepping method is applied firstly to the attitude control of HFV in Ref. [5], and the control design is completed, and the stable tracking of the system is realized. The problem of “differential explosion” occurs easily in the design of the backstepping controller due to the high order of the altitude subsystem of HFV. To solve the problem, the command filter [6], dynamic surface control [8, 9], tracking differentiator [10, 11], and other technologies are widely used in the backstepping control method. To deal with uncertainties as well as external disturbance of HFV, disturbance observers [12, 13] and intelligent approximations [14–16] are widely used in controller design. In addition, in order to improve the transient performance of the control system, the prescribed performance method is applied in the controller design [17–19]. This method originally proposed by Bechlioulis and Rovithakis [20], which could characterize the convergence rate and maximum overshoot of tracking error such that the desired transient performance is achievable by limiting the tracking error in the prescribed convergence range.Although a lot of progress has been made in the controller design of HFV, the AOA constraint has not gotten enough attention. The scramjet engine used in HFV is extremely sensitive to AOA, and its intake efficiency depends on the AOA. The amplitude of AOA must meet certain constraints to ensure the normal operation of the scramjet, otherwise it will cause scramjet thermal choke problem [21]. The traditional method that makes AOA meet the constraint requirements is to make AOA converge to a desired trim value [22, 23]. However, this method cannot prove that the AOA would be guaranteed to fall into predefined interval in theory. At present, barrier Lyapunov function (BLF) has been effectively used in dealing with the problem of AOA constraint [24–29]. The main characteristic of BLF is that its value tends to infinity as the system approaches a predefined boundary. The boundedness of the BLF indicates that the AOA error is constrained within the specified range. Ref. [24] proposes a modified adaptive dynamic surface control scheme. An asymmetric BLF is introduced in this scheme to guarantee state constraints of HFV. However, in the above literatures on the AOA constraint, the proposed control schemes need to assume that the initial error of AOA satisfies the constraint condition. In practice, the initial error of AOA is difficult to obtain accurately. In addition, most studies focus on the steady-state performance of AOA error and ignore its dynamic performance. If the tracking error has a good dynamic performance, the AOA constraint problem will be better handled and the maneuverability of HFV will be further improved.In practice, the control force provided by the actuator is limited. It is easy to cause the problem of actuator saturation due to the high-altitude flight of HFV and the influence of the external environment [30]. The ideal control law cannot be effectively implemented when the system is saturated, which leads to large deviation in command tracking and even seriously affecting the stability of the system [31]. Therefore, the input saturation problem cannot be ignored. The commonly method is to construct the auxiliary system in the controller design process [31–35]. The auxiliary variables in the system compensate the tracking error and stabilize the system when actuators are saturated. However, the existence of auxiliary variables affects the convergence of tracking errors inevitably. Therefore, Refs. [33, 34] construct new auxiliary system whose auxiliary variable can converge with faster speed and higher precision when actuator exits saturation. However, the above studies only consider the amplitude saturation of the control input. In practice, the rate of actuators of HFV has also certain limitations. Ref. [36] further considers the saturation of the actuator’s amplitude and rate at the same time. The command filter is constructed to constrain the control input, but this method lacks theoretical proof, and the constraint effect of filter is difficult to guarantee in [36]. Ref. [37] proposes an adaptive control scheme with multiple constraints on the amplitude and rate of the actuator. Although the amplitude and rate of the control input are constrained effectively by this scheme, the controller design is too complicated.In general, in order to solve the control problem of HFV considering AOA limitation and actuator constraint of amplitude and rate, an antisaturation backstepping control strategy based on the constrained command filter is proposed. Compared with the existing research work, the main contributions of this paper are as follows:(1)
A control method combining the constrained command filter and prescribed performance method is proposed to handle the problem of AOA constraint. Compared with Refs. [24–29], there is no need to assume that the initial error of AOA is within the limited range in this method, which is easier to implement in practice. In addition, the transient performance of AOA error is further improved, and the theoretical proof that AOA meets the constraint conditions is completed(2)
A control method combining constrained command filter and auxiliary system is proposed to handle actuator constraint. Different from Refs. [31–35], this paper further considers the saturation of the actuator’s rate. The control input can satisfy the limit conditions of amplitude and rate in this paper. Compared with Ref. [36, 37], the control law design in this paper is simpler and easier to implement in practice, and the theoretical proof is completed to guarantee the effectiveness of proposed methodThe rest of this paper is organized as follows: Section2 formulates the HFV model and preliminaries; Section 3 presents the controller design process; Sections 4 and 5 give a stability analysis and a simulation study, respectively; the conclusions are proposed in Section 6.


## 2. HFV Model and Preliminaries

### 2.1. HFV Model

In this paper, we consider the longitudinal control-oriented FHV model developed by Ref. [2]:
(1)V̇=Tcosα−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+TsinαmV−gcosγV,(4)α̇=q−γ̇,(5)q̇=MIyy,where velocity V, altitude h, flight-path γ, AOA α, and pitch rate q are rigid body states; m, g, and Iyy represent vehicle mass, acceleration owing to gravity, and moment of inertia, respectively; and T, D, L, and M represent thrust, drag force, lift, and pitch moment, respectively, with the following expressions [18]:
(6)D≈QSCDα2α2+CDαα+CD0,L=L0+Lαα,T=TΦαΦ+T0α,M=MT+M0α+Mδeδe,where Q=1/2ρV2 represents dynamic pressure, in which ρ is air density; S represents reference area; Φ, δe represent deflection fuel equivalency ratio and elevator angular deflection, respectively; CDα2, CDα, CD0, L0, Lα, TΦα, T0α represent the related aerodynamic parameters of thrust, drag force, and lift, respectively; and MT, M0α., and Mδe represent the related parameters of pitch moment.Assumption 1.
The value of the termTsinα in equation (3) is much smaller than the value of lift L, so this term can be ignored.The assumption can be verified by calculation according to the model data and the range of states provided in Ref. [2]. In addition, the rationality of the assumption is also explained and analyzed in Ref. [38].The outputs of the system model are velocityV and altitude h; the control inputs are fuel equivalent ratio Φ and elevator angular deflection δe. According to the models (1)–(5) and Assumption 1, it can be seen that the velocity V change is mainly controlled by the fuel equivalent ratio Φ, and the elevator angular deflection δe controls the altitude h by directly controlling the change of the pitch rate q and then controlling the change of AOA α and flight-path γ. In order to facilitate the design of the control law, the models (1)–(5) can be decomposed into velocity subsystem and height subsystem under normal circumstances [39]:
(7)V̇=fV+gVΦ+d1,(8)ḣ=Vsinγ,γ̇=fγ+gγα+d2,α̇=fα+q−d2,q̇=fq+gqδe+d3,where
(9)fV=T0αcosα−Dm−gsinγ,gV=TΦαcosαm,fγ=L0−mgcosγmV,gγ=LαmV,fα=−fγ−gγα,fq=MT+M0αIyy,gq=MδeIyy.dii=1,2,3 represent disturbance items, including external interference and parameter perturbation, and the following assumption is made:Assumption 2.
The first derivatives of disturbance termsdii=1,2,3 are bounded.Assumption2 is common in existing results for the longitudinal motion model of HFV, and the analysis of literatures has shown that this assumption is in line with the actual physical system and external flight environment of HFV, such as Refs. [24, 33].


### 2.2. Actuator Saturation and AOA Constraint

In order to avoid the phenomenon of thermal choke in the actual project, the fuel equivalent ratioΦ needs to be within a certain range, otherwise scramjet will stop working [34]. The restricted fuel equivalent ratio Φ can be described as
(10)Φ=HΦΦd=Φmax,Φd≥Φmax,Φd,Φmin<Φd<ΦmaxΦmin,Φd≤Φmin,,where Φd represents the ideal control law and Φmax and Φmin are the upper and lower bounds of Φ, respectively.The amplitude and rate of the elevator angular deflectionδe are limited due to the deflection limit of actual physical mechanism, and the limited conditions can be described as
(11)δe=Hδeδed=δmax,δed≥δmax,δed,δmin<δed<δmax,δmin,δed≤δmin,(12)Ψe=HΨΨed=Ψmax,Ψed≥Ψmax,Ψed,Ψmin<Ψed<ΨmaxΨmin,Ψed≤Ψmin,,where δed represents the ideal control law; δmax and δmin are the upper and lower bounds of δe, respectively; Ψe=δ̇e is the rate of elevator angular deflection; Ψed=δ̇ed is the rate of the ideal control law; and Ψmax and Ψmin are the upper and lower bounds of Ψe, respectively.The scramjet of HFV is extremely sensitive to the AOA which directly affects its operating conditions [21]. The normal operation of the scramjet requires AOA to be in an allowable range. So it is necessary to consider AOA constraint which can be expressed as
(13)Am≤α≤AM,where AM and Am are the upper and lower bounds of AOA, respectively.


### 2.3. Prescribed Performance

The prescribed performance [20] method includes performance function and error transformation, which means that while the tracking error converges in an arbitrary small set, the convergence rate and overshoot of the tracking error meet the prescribed conditions.Performance functionϖt is a strictly decreasing positive function. Then, the objective of guaranteeing prescribed tracking performance is equivalent to ensure that
(14)−ϖt<et<ϖt.To transform inequality constraints into equality constraints, define(15)et=ϖtSε,where ε denotes transformed error. The conversion function is defined as Sε=eε−e−ε/eε+e−ε. It is easy to know that Sε is a smooth and strictly increasing reversible function with the following properties:
(16)−1<Sε<1,limε⟶−∞Sε=−1,limε⟶∞Sε=1.Further, the inverse transformation ofSε is defined as
(17)ε=S−1e,ϖ=12ln1+e/ϖ1−e/ϖ.Remark 3.
According to equation (17), if ε is bounded, then inequality (14) holds. That is, the tracking error of system is not only bounded but also limited within the setting range, which guarantees that it meets the requirements of prescribed transient and steady-state performance.


### 2.4. Command Filter

The command filter in [40] is as follows:
(18)χ̇1=χ2,χ̇2=2τωω22τωxd−χ1−χ2,where xd is the filter input; the outputs χ1 and χ2 are the estimated value of xd and its first derivative; the filter design parameters are ω>0 and τ∈0,1.According to Ref. [40], there are limω⟶∞χ1=xd and limω⟶∞χ2=ẋd. Therefore, the following assumption is made.Assumption 4.
There exist unknown constantsη1>0, η2>0 such that x1−xd≤η1 and x2−ẋd≤η2.


### 2.5. Linear Extended State Observer

Consider the following first-order uncertain system:(19)ẋ1t=fx1+gx1u+dt,y=x1t,where dt represents the uncertain term, it is assumed that dt is continuous, and its first derivative is bounded. For this system, LESO can be established as shown below:
(20)e1t=x1t−x^1t,x^̇1t=x^2t+l1e1t+fx1+gx1u,x^̇2t=l2e1t,where x^1t is the estimation value of x1t; x^2t is the estimation value of dt; and li>0i=1,2.The bandwidth configuration method [41] is used to select the parameters which meet the following conditions: l1,l2=ω0α1,ω0α2, where ω0 is the bandwidth of the observer, and the parameters are chosen as αi=3!/i!∙3−i!i=1,2. According to Ref. [42] on the proof of LESO convergence that the observation error d~=x^2−d converges to zero in a finite time, the assumption can be made as follows.Assumption 5.
LESO estimation error is bounded, and there exists an unknown constantD~>0, such that d~≤D~.


## 2.1. HFV Model

In this paper, we consider the longitudinal control-oriented FHV model developed by Ref. [2]:
(1)V̇=Tcosα−Dm−gsinγ,(2)ḣ=Vsinγ,(3)γ̇=L+TsinαmV−gcosγV,(4)α̇=q−γ̇,(5)q̇=MIyy,where velocity V, altitude h, flight-path γ, AOA α, and pitch rate q are rigid body states; m, g, and Iyy represent vehicle mass, acceleration owing to gravity, and moment of inertia, respectively; and T, D, L, and M represent thrust, drag force, lift, and pitch moment, respectively, with the following expressions [18]:
(6)D≈QSCDα2α2+CDαα+CD0,L=L0+Lαα,T=TΦαΦ+T0α,M=MT+M0α+Mδeδe,where Q=1/2ρV2 represents dynamic pressure, in which ρ is air density; S represents reference area; Φ, δe represent deflection fuel equivalency ratio and elevator angular deflection, respectively; CDα2, CDα, CD0, L0, Lα, TΦα, T0α represent the related aerodynamic parameters of thrust, drag force, and lift, respectively; and MT, M0α., and Mδe represent the related parameters of pitch moment.Assumption 1.
The value of the termTsinα in equation (3) is much smaller than the value of lift L, so this term can be ignored.The assumption can be verified by calculation according to the model data and the range of states provided in Ref. [2]. In addition, the rationality of the assumption is also explained and analyzed in Ref. [38].The outputs of the system model are velocityV and altitude h; the control inputs are fuel equivalent ratio Φ and elevator angular deflection δe. According to the models (1)–(5) and Assumption 1, it can be seen that the velocity V change is mainly controlled by the fuel equivalent ratio Φ, and the elevator angular deflection δe controls the altitude h by directly controlling the change of the pitch rate q and then controlling the change of AOA α and flight-path γ. In order to facilitate the design of the control law, the models (1)–(5) can be decomposed into velocity subsystem and height subsystem under normal circumstances [39]:
(7)V̇=fV+gVΦ+d1,(8)ḣ=Vsinγ,γ̇=fγ+gγα+d2,α̇=fα+q−d2,q̇=fq+gqδe+d3,where
(9)fV=T0αcosα−Dm−gsinγ,gV=TΦαcosαm,fγ=L0−mgcosγmV,gγ=LαmV,fα=−fγ−gγα,fq=MT+M0αIyy,gq=MδeIyy.dii=1,2,3 represent disturbance items, including external interference and parameter perturbation, and the following assumption is made:Assumption 2.
The first derivatives of disturbance termsdii=1,2,3 are bounded.Assumption2 is common in existing results for the longitudinal motion model of HFV, and the analysis of literatures has shown that this assumption is in line with the actual physical system and external flight environment of HFV, such as Refs. [24, 33].


## 2.2. Actuator Saturation and AOA Constraint

In order to avoid the phenomenon of thermal choke in the actual project, the fuel equivalent ratioΦ needs to be within a certain range, otherwise scramjet will stop working [34]. The restricted fuel equivalent ratio Φ can be described as
(10)Φ=HΦΦd=Φmax,Φd≥Φmax,Φd,Φmin<Φd<ΦmaxΦmin,Φd≤Φmin,,where Φd represents the ideal control law and Φmax and Φmin are the upper and lower bounds of Φ, respectively.The amplitude and rate of the elevator angular deflectionδe are limited due to the deflection limit of actual physical mechanism, and the limited conditions can be described as
(11)δe=Hδeδed=δmax,δed≥δmax,δed,δmin<δed<δmax,δmin,δed≤δmin,(12)Ψe=HΨΨed=Ψmax,Ψed≥Ψmax,Ψed,Ψmin<Ψed<ΨmaxΨmin,Ψed≤Ψmin,,where δed represents the ideal control law; δmax and δmin are the upper and lower bounds of δe, respectively; Ψe=δ̇e is the rate of elevator angular deflection; Ψed=δ̇ed is the rate of the ideal control law; and Ψmax and Ψmin are the upper and lower bounds of Ψe, respectively.The scramjet of HFV is extremely sensitive to the AOA which directly affects its operating conditions [21]. The normal operation of the scramjet requires AOA to be in an allowable range. So it is necessary to consider AOA constraint which can be expressed as
(13)Am≤α≤AM,where AM and Am are the upper and lower bounds of AOA, respectively.


## 2.3. Prescribed Performance

The prescribed performance [20] method includes performance function and error transformation, which means that while the tracking error converges in an arbitrary small set, the convergence rate and overshoot of the tracking error meet the prescribed conditions.Performance functionϖt is a strictly decreasing positive function. Then, the objective of guaranteeing prescribed tracking performance is equivalent to ensure that
(14)−ϖt<et<ϖt.To transform inequality constraints into equality constraints, define(15)et=ϖtSε,where ε denotes transformed error. The conversion function is defined as Sε=eε−e−ε/eε+e−ε. It is easy to know that Sε is a smooth and strictly increasing reversible function with the following properties:
(16)−1<Sε<1,limε⟶−∞Sε=−1,limε⟶∞Sε=1.Further, the inverse transformation ofSε is defined as
(17)ε=S−1e,ϖ=12ln1+e/ϖ1−e/ϖ.Remark 3.
According to equation (17), if ε is bounded, then inequality (14) holds. That is, the tracking error of system is not only bounded but also limited within the setting range, which guarantees that it meets the requirements of prescribed transient and steady-state performance.


## 2.4. Command Filter

The command filter in [40] is as follows:
(18)χ̇1=χ2,χ̇2=2τωω22τωxd−χ1−χ2,where xd is the filter input; the outputs χ1 and χ2 are the estimated value of xd and its first derivative; the filter design parameters are ω>0 and τ∈0,1.According to Ref. [40], there are limω⟶∞χ1=xd and limω⟶∞χ2=ẋd. Therefore, the following assumption is made.Assumption 4.
There exist unknown constantsη1>0, η2>0 such that x1−xd≤η1 and x2−ẋd≤η2.


## 2.5. Linear Extended State Observer

Consider the following first-order uncertain system:(19)ẋ1t=fx1+gx1u+dt,y=x1t,where dt represents the uncertain term, it is assumed that dt is continuous, and its first derivative is bounded. For this system, LESO can be established as shown below:
(20)e1t=x1t−x^1t,x^̇1t=x^2t+l1e1t+fx1+gx1u,x^̇2t=l2e1t,where x^1t is the estimation value of x1t; x^2t is the estimation value of dt; and li>0i=1,2.The bandwidth configuration method [41] is used to select the parameters which meet the following conditions: l1,l2=ω0α1,ω0α2, where ω0 is the bandwidth of the observer, and the parameters are chosen as αi=3!/i!∙3−i!i=1,2. According to Ref. [42] on the proof of LESO convergence that the observation error d~=x^2−d converges to zero in a finite time, the assumption can be made as follows.Assumption 5.
LESO estimation error is bounded, and there exists an unknown constantD~>0, such that d~≤D~.


## 3. Controller Design

For the control objective, considering that the control system of HFV with AOA constraint and actuator constraints of amplitude and rate, the system output can track the command signal stably, and the actuators and AOA meet the constraints. The structure of control scheme presented for FHV is shown in Figure1.Figure 1
The structure of control scheme presented for FHV.


### 3.1. Velocity Subsystem Controller Design

According to equation (7), the tracking error is defined as e=V−Vd, where Vd is the velocity command. The compensation error is defined as
(21)υV=eV−ξV,where ξV is the auxiliary variable to be designed.Combining equation (7) and deriving equation (21), we get
(22)υ̇V=fV+gVΦ+d1−V̇d−ξ̇V.The subsystem control law is designed as(23)Φd=1gV−kVυV−fV+V̇d−d^1−kVξξV,where kV>0, kVξ>0, and d^1 is the LESO estimation of d1.Considering the input saturation problem of fuel equivalent ratioΦ, by substituting equation (23) into equation (10), the actual control input can be obtained:
(24)Φ=HΦΦd.In order to guarantee the stability of system when fuel equivalent ratioΦ is saturated, the auxiliary compensation system is designed as
(25)ξ̇V=−kVξξV+gVΦ−Φd.Substituting equations (23)–(25) into equation (22), we get
(26)υ̇V=−kVυV+d1−d^1.


### 3.2. Height Subsystem Controller Design

Define the height tracking erroreh=h−hd, where hd is the height command. According to Ref. [24], the virtual control law is designed as
(27)γd=arcsin−kheh+ḣdV,where kh>0 is the designed parameter.Define the errors of flight path, AOA, and pitch rate, respectively(28)eγ=γ−γd,eα=α−αd,eq=q−qd,where αd and qd are the virtual control laws to be designed.Considering that it is difficult to obtain the derivative of the virtual law in the design of the backstepping controller, the command filter is introduced to estimate the virtual commandγd and its derivative γ̇d.
(29)χ̇γ1=χγ2,χ̇γ2=2τγωγωγ22τγωγγd−χγ1−χγ2,where τγ>0 and ωγ>0.Step 1.
Define compensation error(30)υγ=eγ−ξγ,where ξγ is the auxiliary variable to be designed.Combining equation (8) and deriving (30), we get
(31)υ̇γ=fγ+gγαd+gγeα+d2−γ̇d−ξ̇γ.Considering the problem of AOA constraint, the desired control lawαc needs to be limited; and in order to ensure the differentiability of the limiting command, the following constrained command filter is constructed to obtain the control law αd:
(32)χ̇α1=χα2,χ̇α2=2ταωαωα22ταωαsatαc−χα1−χα2,αd=χα1,where τα>0 and ωα>0. The saturation function is as follows:
(33)satαc=αM,αc>αM,αc,αm≤αc≤αMαm,αc<αM,,where αM>0 and αm<0 are the upper and lower bounds of AOA command, respectively.Remark 6.
In equation (32), the initial values of χα1 and χα2 are set as χα10=α0 and χα20=−ωα+1χα10.The purpose is to make the initial error of AOA zero and meet specific constraint. Compared with Refs. [24–29], there is no need to assume that the initial error satisfies the constraints. And it can be ensured that the AOA command αd meets the limiting condition by assigning an initial value; the relevant proof will be given below.In order to offset the influence caused by limiting AOA command, the auxiliary compensation system is designed as(34)ξ̇γ=−kγξξγ+gγsatαc−αc,where kγξ>0.Combining equations (31) and (34), the virtual control law is designed as
(35)αc=1gγ−kγυγ−gγ2υγ2−fγ−d^2+χγ2−kγξξ,where kγ>0 and d^2 is the LESO estimation of d2.Substituting equations (34) and (35) into equation (31), we get
(36)υ̇γ=−kγυγ+gγeα+gγαd−satαc−12gγ2υγ+d2−d^2+χγ2−γ̇d.Step 2.
Define the performance function for AOA error(37)ϖαt=ϖ0−ϖ∞exp−μt+ϖ∞,where ϖ0>0, ϖ∞>0, μ>0,and ϖ0>ϖ∞.According to equation (17), the AOA error is transformed as
(38)εαt=12ln1+eα/ϖα1−eα/ϖα.Remark 7.
This paper constrains the AOA error within the prescribed range by using the prescribed performance method, which can further improve the transient performance of the tracking error while ensuring its steady-state performance.Combining equations (28) and (8), the derivative of εα is given by
(39)ε̇α=rαfα+qd+eq−d2−α̇d+να,where
(40)rα=11−eα/ϖα2∙1ϖα>0,να=11−eα/ϖα2∙ϖ̇αϖα2The virtual control law is designed as(41)qd=−kαεα−ϖα2rαεα−σαrα4εα−fα−d^2+α̇d−ναrα−ξq,where kα>0, σα>0 and ξq is the auxiliary variable to be designed.Substituting equation (41) into equation (39),
(42)ε̇α=−kαrαεα−ϖα2+σαrα24εα+rαeq−ξq−rαd2−d^2.Remark 8.
The solution of the problem of AOA constraint is restricting AOA command and tracking error separately in this paper. That is to ensure thatαm<αd<αM and eα<ϖα0, so that α∈αm−ϖα0,αM+ϖα0.Step 3.
Define compensation error(43)υq=eq−ξq.Combining equations (28) and (8), the derivative of υq is given by
(44)υ̇q=fq+gqδe+d3−q̇d−ξ̇q.Considering the problem of the elevator angular deflectionδe constraints of amplitude and rate, a constrained command filter is constructed to restrain the desired control law δed and obtain the actual control input:
(45)χ̇δ1=χδ2,χ̇δ2=2τδωδHΨωδ22τδωδδed−χδ1−χδ2,δe=Hδχδ1,where τδ>0 and ωδ>0; the specific definitions of Hδ∙ and HΨ∙ are given by equations (10) and (11).Remark 9.
Compared with Ref. [36], the constrained command filter can ensure that the elevator angular deflection δe meets the constraint conditions of amplitude and rate. The specific proof will be given below.In order to offset the influence of input saturation, the auxiliary system is designed:(46)ξ̇q=−kqξ1ξq−kqξ2gqξq+gqδe−δed,where kqξ1>0 and kqξ2>0.Combining equations (44) and (46), the desired control law is designed as
(47)δed=1gq−kqυq−fq−d^3+χq2−rαεα−kqξ1ξq−kqξ2gqξq,where kq>0; d^3 is the LESO estimation of d3; and χq2 is the estimation of the derivative of the virtual control law q̇d, which can be obtained by the following command filter.
(48)χ̇q1=χq2,χ̇q2=2τqωqωq22τqωqqd−χq1−χq2,where τq>0 and ωq>0.Substituting equations (46) and (47) into (44) at the same time, we get
(49)υ̇q=−kqυq+d3−d^3+χq2−q̇d−rαεα.


## 3.1. Velocity Subsystem Controller Design

According to equation (7), the tracking error is defined as e=V−Vd, where Vd is the velocity command. The compensation error is defined as
(21)υV=eV−ξV,where ξV is the auxiliary variable to be designed.Combining equation (7) and deriving equation (21), we get
(22)υ̇V=fV+gVΦ+d1−V̇d−ξ̇V.The subsystem control law is designed as(23)Φd=1gV−kVυV−fV+V̇d−d^1−kVξξV,where kV>0, kVξ>0, and d^1 is the LESO estimation of d1.Considering the input saturation problem of fuel equivalent ratioΦ, by substituting equation (23) into equation (10), the actual control input can be obtained:
(24)Φ=HΦΦd.In order to guarantee the stability of system when fuel equivalent ratioΦ is saturated, the auxiliary compensation system is designed as
(25)ξ̇V=−kVξξV+gVΦ−Φd.Substituting equations (23)–(25) into equation (22), we get
(26)υ̇V=−kVυV+d1−d^1.


## 3.2. Height Subsystem Controller Design

Define the height tracking erroreh=h−hd, where hd is the height command. According to Ref. [24], the virtual control law is designed as
(27)γd=arcsin−kheh+ḣdV,where kh>0 is the designed parameter.Define the errors of flight path, AOA, and pitch rate, respectively(28)eγ=γ−γd,eα=α−αd,eq=q−qd,where αd and qd are the virtual control laws to be designed.Considering that it is difficult to obtain the derivative of the virtual law in the design of the backstepping controller, the command filter is introduced to estimate the virtual commandγd and its derivative γ̇d.
(29)χ̇γ1=χγ2,χ̇γ2=2τγωγωγ22τγωγγd−χγ1−χγ2,where τγ>0 and ωγ>0.Step 1.
Define compensation error(30)υγ=eγ−ξγ,where ξγ is the auxiliary variable to be designed.Combining equation (8) and deriving (30), we get
(31)υ̇γ=fγ+gγαd+gγeα+d2−γ̇d−ξ̇γ.Considering the problem of AOA constraint, the desired control lawαc needs to be limited; and in order to ensure the differentiability of the limiting command, the following constrained command filter is constructed to obtain the control law αd:
(32)χ̇α1=χα2,χ̇α2=2ταωαωα22ταωαsatαc−χα1−χα2,αd=χα1,where τα>0 and ωα>0. The saturation function is as follows:
(33)satαc=αM,αc>αM,αc,αm≤αc≤αMαm,αc<αM,,where αM>0 and αm<0 are the upper and lower bounds of AOA command, respectively.Remark 6.
In equation (32), the initial values of χα1 and χα2 are set as χα10=α0 and χα20=−ωα+1χα10.The purpose is to make the initial error of AOA zero and meet specific constraint. Compared with Refs. [24–29], there is no need to assume that the initial error satisfies the constraints. And it can be ensured that the AOA command αd meets the limiting condition by assigning an initial value; the relevant proof will be given below.In order to offset the influence caused by limiting AOA command, the auxiliary compensation system is designed as(34)ξ̇γ=−kγξξγ+gγsatαc−αc,where kγξ>0.Combining equations (31) and (34), the virtual control law is designed as
(35)αc=1gγ−kγυγ−gγ2υγ2−fγ−d^2+χγ2−kγξξ,where kγ>0 and d^2 is the LESO estimation of d2.Substituting equations (34) and (35) into equation (31), we get
(36)υ̇γ=−kγυγ+gγeα+gγαd−satαc−12gγ2υγ+d2−d^2+χγ2−γ̇d.Step 2.
Define the performance function for AOA error(37)ϖαt=ϖ0−ϖ∞exp−μt+ϖ∞,where ϖ0>0, ϖ∞>0, μ>0,and ϖ0>ϖ∞.According to equation (17), the AOA error is transformed as
(38)εαt=12ln1+eα/ϖα1−eα/ϖα.Remark 7.
This paper constrains the AOA error within the prescribed range by using the prescribed performance method, which can further improve the transient performance of the tracking error while ensuring its steady-state performance.Combining equations (28) and (8), the derivative of εα is given by
(39)ε̇α=rαfα+qd+eq−d2−α̇d+να,where
(40)rα=11−eα/ϖα2∙1ϖα>0,να=11−eα/ϖα2∙ϖ̇αϖα2The virtual control law is designed as(41)qd=−kαεα−ϖα2rαεα−σαrα4εα−fα−d^2+α̇d−ναrα−ξq,where kα>0, σα>0 and ξq is the auxiliary variable to be designed.Substituting equation (41) into equation (39),
(42)ε̇α=−kαrαεα−ϖα2+σαrα24εα+rαeq−ξq−rαd2−d^2.Remark 8.
The solution of the problem of AOA constraint is restricting AOA command and tracking error separately in this paper. That is to ensure thatαm<αd<αM and eα<ϖα0, so that α∈αm−ϖα0,αM+ϖα0.Step 3.
Define compensation error(43)υq=eq−ξq.Combining equations (28) and (8), the derivative of υq is given by
(44)υ̇q=fq+gqδe+d3−q̇d−ξ̇q.Considering the problem of the elevator angular deflectionδe constraints of amplitude and rate, a constrained command filter is constructed to restrain the desired control law δed and obtain the actual control input:
(45)χ̇δ1=χδ2,χ̇δ2=2τδωδHΨωδ22τδωδδed−χδ1−χδ2,δe=Hδχδ1,where τδ>0 and ωδ>0; the specific definitions of Hδ∙ and HΨ∙ are given by equations (10) and (11).Remark 9.
Compared with Ref. [36], the constrained command filter can ensure that the elevator angular deflection δe meets the constraint conditions of amplitude and rate. The specific proof will be given below.In order to offset the influence of input saturation, the auxiliary system is designed:(46)ξ̇q=−kqξ1ξq−kqξ2gqξq+gqδe−δed,where kqξ1>0 and kqξ2>0.Combining equations (44) and (46), the desired control law is designed as
(47)δed=1gq−kqυq−fq−d^3+χq2−rαεα−kqξ1ξq−kqξ2gqξq,where kq>0; d^3 is the LESO estimation of d3; and χq2 is the estimation of the derivative of the virtual control law q̇d, which can be obtained by the following command filter.
(48)χ̇q1=χq2,χ̇q2=2τqωqωq22τqωqqd−χq1−χq2,where τq>0 and ωq>0.Substituting equations (46) and (47) into (44) at the same time, we get
(49)υ̇q=−kqυq+d3−d^3+χq2−q̇d−rαεα.


## 4. Stability Analysis

Theorem 10.
For the system (7) and (8), by using equations (24) and (47) to limit the control input, it is guaranteed that fuel equivalent ratio Φ and elevator angular deflection δe meet the restricted conditions, respectively, such that Φ∈Φmin,Φmax, δe∈δmin,δmax, and Ψe=δ̇e∈Ψmin,Ψmax.Proof.
SinceΦ=HΦΦd and δe=Hδχδ1, it is easy to get Φ∈Φmin,Φmax and δe∈δmin,δmax according to the definition of saturation function HΦ∙ and Hδ∙.
The term(50)χ̇δ2=2τδωδHΨωδ22τδωδδed−χδ1−χδ2in equation (45) is transformed as
(51)χ̇δ2+cδχδ2=cδHΨωδ2cδδed−χδ1,where cδ=2τδωδ. Due to the saturation function HΨ∙∈Ψmin,Ψmax, equation (51) is satisfied with
(52)cδΨmin≤χ̇δ2+cδχδ2≤cδΨmax.
Multiplying inequality (52) by expcδt, we have
(53)cδΨminexpcδt≤χδ2expcδt′≤cδΨmaxexpcδt.
Integrating inequality (53) yields
(54)χδ2≥Ψmin1−exp−cδt+χδ20,χδ2≤Ψmax1−exp−cδt+χδ20.
In practice, the upper and lower bounds ofΨe satisfy Ψmin<0 and Ψmax>0. Taking the initial value χδ20=0, equation (54) can be simplified to
(55)Ψmin≤χδ2≤Ψmax.
Whenχδ1∈δmin,δmax, then δe=Hδχδ1=χδ1, and Ψe=δ̇e∈Ψmin,Ψmax; when χδ1∉δmin,δmax, then δet=δmin or δet=δmax, such that δ̇e=0∈Ψmin,Ψmax, so Ψe=δ̇e∈Ψmin,Ψmax.☐Remark 11.
The constrained command filter in Ref. [36] constructed for limiting the amplitude and rate of control input is designed as
(56)χ̇δ1=χδ2,χ̇δ2=2τδωδHΨωδ22τδωδHδδed−χδ1−χδ2,δe=χδ1.If the functionHΨ∙ reaches the saturation point, equation (56) is transformed as
(57)χ̇δ1=χδ2,χ̇δ2=2τδωδΨm−χδ2,δe=χδ1,where Ψm=Ψmin or Ψm=Ψmax. Obviously, equation (57) cannot ensure that the amplitude of δe meets the restricted conditions. Therefore, the constrained command filter constructed in Ref. [36] cannot guarantee that it achieves effective constraint on δe.Theorem 12.
For the system (7) and (8), under Assumptions 1–5, consider the constraints of AOA and actuators, and use designed control laws (24) and (47).
(1)
The tracking error of the closed-loop system is ultimately uniformly bounded(2)
AOA satisfiesα∈αm−ϖα0,αM+ϖα0Proof.
Define Lyapunov function for system as(58)W=12υV2+υγ2+εα2+υq2.
Combining equations (26), (36), (42), and (49), the derivative ofWis obtained as
(59)Ẇ=υV−kVυV+d~1+υγ−kγυγ+gγeα+gγηα−12gγ2υγ+d~2+η̇γ+εα−kαrαεα−ϖα2+σαrα24εα+rαυq−rαd~2+υq−kqυq+d~3+η̇q−rαεα,where ηγ=χγ1−γd, ηα=αd−satαc, and ηq=χq1−qd are the errors of command filter and d~i=di−d^ii=1,2,3 are the estimation errors of LESO.☐According to Assumption4, there exist unknown constantsNii=1,2,3>0, such that η̇γ≤N1, ηα≤N2, and η̇q≤N3. According to Assumption 5, there exist unknown constants D~ii=1,2,3>0, such that d~1≤D~1, d~2≤D~2, and d~3≤D~3.The following inequalities in equation (59) hold:
(60)υVd~1≤kV2υV2+12kVd~12,υγgγeα≤14gγ2υγ2+eα2,υγgγηα≤14gγ2υγ2+ηα2,υγd~2≤kγ4υγ2+1kγd~22,υγη̇γ≤kγ4υγ2+1kγη̇γ2,−rαεαd~2≤σα4rα2εα2+1σαd~22,υqd~3≤kq4υq2+1kqd~32,υqη̇q≤kq4υq2+1kqη̇q2.Equation (38) can be transformed as
(61)tanhεα=eαϖα.According to the mean value theorem,tanhεα=1−tanh2εαεα=εα′∙εα, where εα′ is an unknown constant which lies on the closed interval formed by 0 and εα. Note that 1−tanh2εα≤1; equation (61) satisfies
(62)εαϖα2≤eα2.Combining equations (60) and (62), equation (59) can be simplified to
(63)Ẇ≤−kV2υV2−kγ2υγ2−rαkαεα2−kq2υq2+12kVd~12+1kγ+1σαd~22+1kqd~32+ηα2+1kγη̇γ2+1kqη̇q2.LetΛ=υV,υγ,εα,υqT and k¯=minkV/2,kγ/2,rαkα,kq/2T. According to the previous analysis, there exists an unknown constant NW>0, such that
(64)12kVd~12+1kγ+1σαd~22+1kqd~32+ηα2+1kγη̇γ2+1kqη̇q2≤NW.Furthermore, equation (63) is obtained as
(65)Ẇ≤−k¯Λ22+NW.WhenΛ≥NW/k¯, equation (65) is satisfied with
(66)Ẇ≤0.Therefore, it can be stated thatW is bounded and according to the definition of W. υV,υγ,εα, and υq are bounded. According to Remark 3, eα is bounded and meets the prescribed conditions by the boundedness of εα.When the control inputs exit saturation and the ideal AOA command satisfies the constraint conditions, there areΦ−Φd=0 and satαc−αc=0 for the auxiliary systems (25) and (34), then the auxiliary variables ξV⟶0 and ξγ⟶0, such that υV⟶eV, υγ⟶eγ. Therefore, eV and eγ are bounded obviously.According to the auxiliary system (46), we choose the Lyapunov function Wξ=1/2ξq2 and its derivative is obtained as
(67)Ẇξ=−kqξ1ξq2−kqξ2gqξq2+gqδe−δedξq≤−kqξ1ξq2−kqξ2gqξq2−δe−δedξqkqξ2.When the control input exits saturation,δed∈δmin,δmax, notice that limωδ⟶∞χδ1=δed, and δe=Hδχδ1, so in equation (67), there is δe−δed∈l∞. When the auxiliary variable satisfies ξq≥δe−δed/kqξ2, equation (67) can be simplified to Ẇξ≤−kqξ1ξq2 obviously. Therefore, ξq is bounded and eq is bounded.Equation (32) is transformed as
(68)χ¨α1+2ταωαχ̇α1+ωα2χα1=ωα2satαc.Sinceαm≤satαc≤αM, equation (68) is satisfied with
(69)αmωα2≤χ¨α1+2ταωαχ̇α1+ωα2χα1≤αMωα2.Letτα=1; multiplying inequality (69) by expϖαt, we get
(70)αmωα2expϖαt≤χα1expϖαt′′≤αMωα2expϖαt.Integrating inequality (70), we get
(71)χα1≥αm1−e−ϖαt1+ϖαt+1+ϖαχα1+χ̇α10,χα1≤αM1−e−ϖαt1+ϖαt+1+ϖαχα1+χ̇α10.According to the initial assignment of Remark6, 1+ϖαχα1+χ̇α10=0, since αM>0, αm<0, and αd=χα1, equation (71) can be simplified to
(72)αm≤αd≤αM.In addition, according to Theorem10, it can be obtained eα<ϖαt by the boundedness of εα, and furthermore,
(73)−ϖα0<eα<ϖα0.According to equations (72) and (73) and α=eα+αd, we have
(74)α∈αm−ϖα0,αM+ϖα0.


## 5. Simulation Results

In order to verify the effectiveness of the control scheme proposed in this paper, the longitudinal motion model (1)–(5) of HFV is taken as the object, and the controller designed in Section 3 is used to perform MATLAB simulation. The relevant parameters of the HFV model are taken from Ref. [2].The gains of the controller are chosen askV=0.4, kh=0.5, kγ=0.2, kα=1, kq=2, and σα=0.001. The parameters of the constrained command filters are chosen as τα=1, τγ=τq=0.8, τδ=0.5, ωγ=ωα=10, ωq=20, and ωδ=90. The parameters of the prescribed performance are chosen as μ=0.1, ϖ0=0.01, and ϖ∞=0.005. The parameters of the auxiliary system are chosen as kVξ=0.1, kγξ=0.2, kqξ1=0.2, and kqξ2=0.02. The parameters of LESO are chosen as ωV0=ωγ0=ωq0=5. The external disturbances in d1, d2, and d3 are set as 2sin0.1t, 0.001sin0.1t, and 0.2sin0.1t, respectively. We suppose that there is the following perturbation of 30% in the HFV model
(75)C=C01+0.3sin0.1t,where C represents the simulation value and C0 represents the nominal value.The initial value of the system states is set asV0=7702ft/s, h0=85000ft, γ0=0deg, α0=1.5153deg, and q0=0deg/s. We assume the constraint of AOA is set as α∈−6°,6°, and the boundaries of saturation function (33) are set as
(76)αM=6π/180−ϖα0=0.0947rad,αm=−6π/180+ϖα0=−0.0947rad.The actuator constraints are set asΦ∈0.05,1.5, δe∈−30°,30°, and δ̇e=Ψe∈−100°/s,100°/s. The reference commands Vd,hd are generated, respectively, by passing step commands △Vd=2000ft/s and
(77)△hd=10000ft,0<t≤200s,−5000ft,t>200s,through the following filter command signal [43]:
(78)Gs=0.042s2+0.064s+0.042.In order to verify the superiority of the proposed control scheme (denoted as “A”), the adaptive antisaturation control scheme (denoted as “B”) designed in [20] is introduced for comparison and simulation. For the fairness of the comparison, scheme B selects the same control gain parameters as scheme A and adopts LESO with the same parameters to observe the disturbances.The simulation results are shown in Figures2–13. It can be seen from Figures 2–7 that both control schemes A and B can guarantee that the system tracks the command signal. It can be seen from Figure 2 that the curves of velocity and its tracking error of scheme A are basically the same as scheme B, because both schemes A and B use the dynamic inverse method in the controller design of velocity subsystem. It can be seen from Figures 3–7 that the tracking accuracy of scheme A is better than scheme B obviously for the altitude subsystem. Figure 6 shows that scheme B lacks the limit of the AOA command, which leads to AOA that can achieve command tracking but exceed the constraint range. And it is obvious that AOA of scheme A can track the limiting command stably, and it is always within the constraint range. Figure 7 shows that the constraint range of scheme B is large which affects the tracking accuracy of AOA, because the range needs to be greater than the initial absolute value of AOA error. And the constraint range is small because the initial value of AOA error is zero in scheme A, which makes AOA error converge to zero quickly. It is the initial value of AOA error that affects the setting of error constraint range, which influences the tracking command effect of AOA and even the whole altitude subsystem (see Figures 3–7).Remark 13.
The solution of the AOA constraint problem by scheme A is mainly due to the following two points. Firstly, there is no need to consider the influence of the initial error on the constraint conditions. So the tracking error converges quickly to zero and meets the prescribed transient and steady-state performance (see Figure7). Secondly, the AOA command of scheme A is obtained by the constrained command filter, and its amplitude is within the constraint range (see Figure 8). The effectiveness of the constrained command filter is verified.Figure 2
Curves of velocity and its tracking error.Figure 3
Curves of height and its tracking error.Figure 4
Curves of fight-path and its tracking error.Figure 5
Curves of pitch rate and its tracking error.Figure 6
Tracking curves of AOA.Figure 7
Tracking curves of AOA error.Figure 8
Curves of limited AOA command of scheme A.It can be seen from Figures9–11 that the amplitude and rate of the control input meet the constraint requirements in scheme A. Although scheme B restricts the amplitude of the control input and the actuator meets the requirements, the rate limitation of the actuator is ignored, and the rate of elevator angular deflection cannot meet the constraint conditions. Figure 12 shows that when the control input is saturated or the desired AOA command exceeds the constraint range, the auxiliary variable responds quickly to compensate the tracking error, which ensures the stability of the system. And the auxiliary variable quickly converges to zero when the system exits saturation. Figure 13 shows that LESO can observe system disturbances effectively, and its estimate is bounded.Remark 14.
We can clearly see that the amplitude and rate of the control input meet the constraint requirements in scheme A. The effectiveness of the constrained command filter limiting the control input is verified. And the problem of actuator constraints is solved effectively.Figure 9
Curves of fuel equivalent ratio.Figure 10
Curves of elevator angular deflection.Figure 11
Curves of rate of elevator angular deflection.Figure 12
Curves of auxiliary variable.Figure 13
Curves of LESO observation.In summary, the problem of AOA and actuator constraints taking into account parameter perturbation and external disturbance can be solved successfully by the control scheme designed in this paper. And the system achieves a better command tracking effect. The effectiveness of the proposed control scheme is verified by this simulation result.


## 6. Conclusions

An antisaturation backstepping control scheme based on the constrained command filter is proposed for HFV. The AOA constraint problem is solved by combining the proposed constraint command filter and prescribed performance method. AOA meets the constraint during the process of tracking command by constraining the AOA command and tracking error separately. Then, the amplitude and rate of the actuators meet the restricted requirements due to the constrained command filter constructed for the limited input. The problem of AOA and actuators constraint can be solved effectively by the proposed scheme based on the constrained command filter in this paper through theoretical proof and simulation test. In the future research, the methods for controller design of HFV under states constraint and actuator faults will be investigated. Meanwhile, intelligence method will be added in the proposed control scheme.



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