# Disturbance Observer-Based Adaptive Control of Hypersonic Vehicles with Constrained Actuators

Authors: Guan Wang; Li Li; Weihua Li; Huajun Zhou; Changbo Ma; Hongwei Xia
Journal: International Journal of Aerospace Engineering
 (2020)

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## Abstract

This study investigates an adaptive controller for the flexible air-breathing hypersonic vehicles (AHVs) subject to external disturbances and actuator constraints. The combination of nonlinear disturbance observer and adaptive mechanism is exploited to design an adaptive controller for each subsystem. For the velocity subsystem, an auxiliary system is employed to handle the scramjet input saturation issue. For the altitude subsystem, the magnitude/rate constraints and the dynamics of aerodynamic control surfaces are addressed by the control allocation module. Simulations show the effectiveness of the proposed control.


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## Body

## 1. Introduction

Air-breathing hypersonic vehicles (AHVs) have attracted considerable attention in both military and civil fields. Though AHVs have been developed for more than half a century, various studies have shown that some critical technologies, such as thermal protection, high-fidelity modeling, and robust control, still need to be further investigated.Based on the control-oriented model developed by Parker et al. [1], numerous theoretical studies on the control of AHVs have sprung up [2]. Early research interests focus on state feedback linearization or nonlinear dynamic inverse-based control of AHVs [3]. In recent years, the adaptive control has always been a research hotspot in the field of AHV control, and one can refer to the robust adaptive control in [4, 5], the sliding mode adaptive control in [6], the trajectory restricted adaptive control in [7], the fault-tolerant adaptive control in [8, 9], and the neural adaptive control in [10, 11].Nevertheless, a practical problem in AHV control is the physically limited actuators [12, 13]. Recently, some outstanding works have come out focusing on the input saturation problem of AHVs [14, 15]. [16] used the disturbance observer-based (DOB) feedback linearization control to overcome constraints on actuators. A combination of back-stepping design and nonlinear disturbance observer (NDO) is utilized to handle actuator constraints [17]. By introducing the sliding-mode differentiator techniques and the auxiliary systems, [11] developed the distributed finite time fault-tolerant containment control to handle the input saturation. On the other side, control allocation (CA) also plays a good role in dealing with the actuator constraints [18]. For instance, [19] handled the saturation using a robust model predictive control algorithm. [20] developed a control allocation scheme for fault-tolerant control based on a linear parameter varying (LPV) system. But the employed LPV model was complex and time-demanding for the CA problem.Motivated by the above investigations, this paper proposes a disturbance observer-based adaptive control AHVs with constrained actuators. In the velocity loop, the auxiliary system developed in [12] is employed to handle the saturation issue of the scramjet input. In the altitude loop, a control allocation module is exploited for the magnitude/rate constraints and the dynamics of aerodynamic control surfaces. The main contributions of this paper can be briefly outlined as follows:
(1)
The saturation problem of the scramjet, as well as the dynamics and magnitude/rate constraints of the control surface, is taken into consideration(2)
NDO is developed in each step to further improve the disturbance rejection property. The observation errors are suppressed by adaptive control lawsThe rest of this paper is organized as follows: Section2 formulates the considered problem; Section 3 presents the design process; Sections 4 and 5 give a stability analysis and a simulation study, respectively; Section 6 concludes this study.


## 2. Problem Statement and Preliminaries

### 2.1. Control-Oriented AHV Model

In this paper, we consider the longitudinal control-oriented AHV model developed by Parker et al. [1]:
(1)V̇=Tcosα−Dm−gsinγ+dV,(2)ḣ=Vsinγ,(3)γ̇=L+TsinαmV−gcosγV+dγ,(4)α̇=Q−γ̇,(5)Q̇=MIyy+∑i=12ψiη¨iIyy+dQ,(6)η¨i=−2ξiωiη̇i−ωi2ηi+Ni+ψi′Q̇,i=1,2.In (1)–(5), velocity V, altitude h, flight path angle (FPA) γ, angle of attack (AoA) α, and pitch rate (PR) Q are system states; g is the acceleration due to gravity; mass m and moment of inertia Iyy are decided by the design and fuel level of AHVs; the disturbances dV, dγ, and dQ are external disturbances with bounded derivatives; ηi is the ith generalized elastic coordinate; ωi and ξi are the natural frequency and the damping ratio for flexible mode ηi, respectively; ψi and ψi′ are coupling coefficients between rigid-body and flexible dynamics; thrust T, lift L, drag D, pitching moment M, and ith generalized force Ni have the following curve-fitted approximations:
(7)T=q¯SCT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0Φ+q¯SCT3α3+CT2α2+CT1α+CT0,(8)L=q¯SCL1α+CL0+CLδeδe+CLδcδc,(9)D=q¯SCD2α2+CD1α+CD0+CDδe2δe2+CDδeδe+CDδc2δc2+CDδcδc,(10)M=zTT+q¯c¯SCM2α2+CM1α+CM0+CMδeδe+CMδcδc,(11)N1=CN12α2+CNi1α+CN00,(12)N2=CN22α2+CN21α+CN20+CN2δeδe+CN2δcδc.In (7)–(10), scramjet fuel equivalency ratio (FER) Φ, canard deflection angle (CDA) δc, and elevator deflection angle (EDA) δe are three system inputs, respectively; q¯ is the dynamic pressure; S is the reference area; zT is the thrust-to-moment coupling coefficient; c¯ is the mean aerodynamic chord; C∗∗ denotes the fitting coefficients of force and moment.


### 2.2. Problem Formulation

From a practical viewpoint, FER should comply with the saturation property:(13)Φ=SatΦc=Φcooling,Φc<Φcooling,Φchoking,Φc>Φchoking,Φc,otherwise,where Φc is the FER command; Φcooling is the constant lower threshold to keep sufficient heat dissipation through active cooling; the upper bound Φchoking is required to prevent the scramjet from thermal choking [12].In engineering, the magnitude and rate constraints of the actuators (canard/elevator) have a great influence on control performances.(14)ML≤δ≤MU,RL≤δ̇≤RU,where δ denote EDA δc or CDA δe; ML<0 and MU>0 are lower and upper bounds of the magnitude; RL<0 and RU>0 are the corresponding rate bounds. It should be pointed out that the upper and lower amplitude limits of a control surface can be asymmetric in reality, that is, the more common case ML≠MU or RL≠RU is also taken into consideration.At the end of this subsection, several lemmas crucial for the following design are given.Lemma 1 (see [21]).
x≤xtanhcx/ε+ε, where x∈ℝ, ε∈ℝ+, and c≈0.2785.Lemma 2 (see [22]).
If a functionW satisfies Ẇ≤−aW+b with a∈ℝ+, b∈ℝ+, it has
(15)Wt≤ba+W0−bae−at.Lemma 3 (see [23]).
Consider a scalar nonlinear system(16)ẋ=Hx+Gxu+d,where x∈ℝ and u∈ℝ are system state and control input, respectively; Hx and Gx are known scalar functions with respect to x. d∈ℝ represents the disturbance and ḋ≤Md, where Md is an unknown positive constant.
NDO is constructed as(17)ṗ=−lxλx+p+Hx+Gxu,lx=∂λx∂x,d^=p+λx,where d^∈ℝ is the estimation of d; λx=Ldx with Ld>0; lx=∂λx/∂x=Ld.
Defining the estimation errord~=d−d^, d~ is governed by
(18)d~̇=ḋ−ṗ−∂λx∂xẋ=−lxd~+ḋ.The estimation error system (18) is bounded-input-bounded output stability, that is, d~ is bounded with bounded ḋ.


## 2.1. Control-Oriented AHV Model

In this paper, we consider the longitudinal control-oriented AHV model developed by Parker et al. [1]:
(1)V̇=Tcosα−Dm−gsinγ+dV,(2)ḣ=Vsinγ,(3)γ̇=L+TsinαmV−gcosγV+dγ,(4)α̇=Q−γ̇,(5)Q̇=MIyy+∑i=12ψiη¨iIyy+dQ,(6)η¨i=−2ξiωiη̇i−ωi2ηi+Ni+ψi′Q̇,i=1,2.In (1)–(5), velocity V, altitude h, flight path angle (FPA) γ, angle of attack (AoA) α, and pitch rate (PR) Q are system states; g is the acceleration due to gravity; mass m and moment of inertia Iyy are decided by the design and fuel level of AHVs; the disturbances dV, dγ, and dQ are external disturbances with bounded derivatives; ηi is the ith generalized elastic coordinate; ωi and ξi are the natural frequency and the damping ratio for flexible mode ηi, respectively; ψi and ψi′ are coupling coefficients between rigid-body and flexible dynamics; thrust T, lift L, drag D, pitching moment M, and ith generalized force Ni have the following curve-fitted approximations:
(7)T=q¯SCT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0Φ+q¯SCT3α3+CT2α2+CT1α+CT0,(8)L=q¯SCL1α+CL0+CLδeδe+CLδcδc,(9)D=q¯SCD2α2+CD1α+CD0+CDδe2δe2+CDδeδe+CDδc2δc2+CDδcδc,(10)M=zTT+q¯c¯SCM2α2+CM1α+CM0+CMδeδe+CMδcδc,(11)N1=CN12α2+CNi1α+CN00,(12)N2=CN22α2+CN21α+CN20+CN2δeδe+CN2δcδc.In (7)–(10), scramjet fuel equivalency ratio (FER) Φ, canard deflection angle (CDA) δc, and elevator deflection angle (EDA) δe are three system inputs, respectively; q¯ is the dynamic pressure; S is the reference area; zT is the thrust-to-moment coupling coefficient; c¯ is the mean aerodynamic chord; C∗∗ denotes the fitting coefficients of force and moment.


## 2.2. Problem Formulation

From a practical viewpoint, FER should comply with the saturation property:(13)Φ=SatΦc=Φcooling,Φc<Φcooling,Φchoking,Φc>Φchoking,Φc,otherwise,where Φc is the FER command; Φcooling is the constant lower threshold to keep sufficient heat dissipation through active cooling; the upper bound Φchoking is required to prevent the scramjet from thermal choking [12].In engineering, the magnitude and rate constraints of the actuators (canard/elevator) have a great influence on control performances.(14)ML≤δ≤MU,RL≤δ̇≤RU,where δ denote EDA δc or CDA δe; ML<0 and MU>0 are lower and upper bounds of the magnitude; RL<0 and RU>0 are the corresponding rate bounds. It should be pointed out that the upper and lower amplitude limits of a control surface can be asymmetric in reality, that is, the more common case ML≠MU or RL≠RU is also taken into consideration.At the end of this subsection, several lemmas crucial for the following design are given.Lemma 1 (see [21]).
x≤xtanhcx/ε+ε, where x∈ℝ, ε∈ℝ+, and c≈0.2785.Lemma 2 (see [22]).
If a functionW satisfies Ẇ≤−aW+b with a∈ℝ+, b∈ℝ+, it has
(15)Wt≤ba+W0−bae−at.Lemma 3 (see [23]).
Consider a scalar nonlinear system(16)ẋ=Hx+Gxu+d,where x∈ℝ and u∈ℝ are system state and control input, respectively; Hx and Gx are known scalar functions with respect to x. d∈ℝ represents the disturbance and ḋ≤Md, where Md is an unknown positive constant.
NDO is constructed as(17)ṗ=−lxλx+p+Hx+Gxu,lx=∂λx∂x,d^=p+λx,where d^∈ℝ is the estimation of d; λx=Ldx with Ld>0; lx=∂λx/∂x=Ld.
Defining the estimation errord~=d−d^, d~ is governed by
(18)d~̇=ḋ−ṗ−∂λx∂xẋ=−lxd~+ḋ.The estimation error system (18) is bounded-input-bounded output stability, that is, d~ is bounded with bounded ḋ.


## 3. Controller Design

### 3.1. Velocity Tracking Control Design

This section presents an adaptive tracking controller for velocity subsystem (1) ensuring V→Vref. The velocity dynamics is first expressed in the following form:
(19)V̇=HV+GVΦ+DV,where
(20)HV=q¯SmCT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0cosα−CD2α2+CD1α+CD0−gsinγ,GV=CT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0cosα,DV=−q¯SCDδe2δe2+CDδeδe+CDδc2δc2+CDδcδc+dV.First, similar to [23], we introduce a modified tracking error to handle the saturated Φ as
(21)eV=V−Vref−χV,where χV is governed by
(22)χ̇V=−kVχV+GVΦ−ΦV,with kV>0. Then, ėV is calculated as
(23)ėV=kVχV+HV+GVΦ+DV−V̇ref.Taking into account the boundedḋV and the finite deflecting velocity of physical control surfaces, ḊV is assumed to be bounded. Hence, the NDO for DV is constructed.
(24)ṗV=−lVλVV+pV+HV+GVΦc,lV=∂λVV∂V,D^V=pV+λVV.Defining estimation error asD~V=DV−D^V, it has
(25)D~̇V=−lVD~V+ḊV.For the fact thatḊV is bounded, D~V is also bounded from Lemma 3, i.e., D~V≤ΘV with ΘV>0.We design the control law(26)Φc=1GV−kVV−Vref−HV+V̇ref−DV−Θ^VtanhceVεV,where D^V is estimated by NDO and Θ^V is the estimate of ΘV, generated by the adaptive law:
(27)Θ^̇V=σV−kV′Θ^V−eVtanhceVεV,where σV>0, kV′>0, and εV>0.


### 3.2. Altitude Tracking Control Design

In the altitude subsystem, control allocation is employed as an attractive technology to handle both canard and elevator servomechanism dynamics and constraints. For convenience,δc0 and δe0 represent the desired control command to be designed, δccom and δecom denote the control command after the process of control allocation, and δc and δe are the real control inputs applied to AHVs with the dynamics δc/δccom=20/s+20 and δe/δecom=20/s+20 [24]. We also define Δδc,ΔδeT=δc−δc0,δe−δe0T for the following design.


#### 3.2.1. Design forh,γ-Dynamics

Recalling the kinematical equation (2), the realization of altitude reference tracking can be transferred to the FPA tracking problem. In this paper, to achieve h→href, the control system steers the AHVs to track the FPA reference:
(28)γref=arcsin−kVh−href+ḣrefV,where href is the smooth altitude reference generated by the traditional second-order command filter; kh>0.Define a time-related function asαf, which decides the final AoA. The FPA dynamics can be calculated as
(29)γ̇=RΓα−αf+Hγ+Gγδc0+Dγ,where
(30)RΓ=q¯SmVCL1+CT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0Φ+CT3αf3+CT2αf2+CT1αf+CT0,Hγ=q¯SmVCL1+CT,Φ3αf3+CT,Φ2αf2+CT,Φ1αf+CT,Φ0Φ+CT3αf3+CT2αf2+CT1αf+CT0αf+q¯SmVCL0−gcosγV,Gγ=q¯SmVCLδc,Dγ=dγ,αf+dγ,δ+dγ,dγ,αf=q¯SmVCT,Φ3α3+CT,Φ2α2+CT,Φ1αΦ+CT3α3+CT2α2+CT1α−CT,Φ3αf3+CT,Φ2αf2+CT,Φ1αfΦ−CT3αf3+CT2αf2+CT1αfsinα,dγ,δ=q¯SmVCLδeδe+CLδcΔδc.Considering the real smallα, the approximate residual dγ,αf is assumed to be bounded. Meanwhile, the unknown disturbance introduced by the above control allocation dγ,δ is bounded considering the amplitude and rate constrained actuators. Thus, the boundedness of the disturbance term Dγ is guaranteed.Furthermore, similar to [4], the virtual AoA control is designed as
(31)αd=αf−eγ,α−αf=α−αd−eγ=eα−eγ.Then,ėγ can be rewritten as
(32)ėγ=RΓeα−eγ+Hγ+Gγδc0+Dγ−γ̇ref.Similarly, the following NDO forDγ is constructed.
(33)ṗγ=−lγλγγ+pγ+RΓeα−eγ+Hγ+Gγδc0,lγ=∂λγγ∂γ,D^γ=pγ+λγγ.Defining estimation error asD~γ=Dγ−D^γ, and
(34)D~̇γ=−lγD~γ+Ḋγ.Further noting the boundeddγ,αf, dγ,δ, and dγ, the boundedness of Dγ is guaranteed. From Lemma 3, D~γ is bounded for the fact that Ḋγ is bounded, i.e., D~γ≤Θγ with Θγ>0.We design the control law to regulate theėγ system as
(35)δc0=1Gγ−kγeγ−RΓeα−Hγ+γ̇ref−Dγ−Θ^γtanhceγεγ,where Θ^γ is governed by the adaptive law
(36)Θ^̇γ=σγ−kγ′Θ^γ+eγtanhceγεγ,with σγ>0, εγ>0, and kγ′>0.


#### 3.2.2. Design forα,Q-Dynamics

For the altitude subsystem, the AoA tracking error dynamics have the relationship(37)ėα=α̇−α̇d=α̇−α̇f+ėγ=Q−γ̇ref−α̇f.DefineeQ=Q−Qd with Qd=−kαeα+γ̇ref+α̇f, and ėα can be rewritten as
(38)ėα=−kαeα+eQ,Q̇d=kα2eα−kαeQ+γ¨ref+α¨f.Then, the PR dynamics is formulated as(39)Q̇=HQ+GQδe0+DQ,where
(40)HQ=zTq¯SIyyCT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0Φ+CT3α3+CT2α2+CT1α+CT0,GQ=q¯c¯SIyyCMδe,DQ=dQ+dQ,δ+∑i=12ψiη¨iIyy,dQ,δ=q¯c¯SIyyCMδcδc+CMδeΔδe.Following a similar analysis to previous steps, the boundedness ofDQ is also guaranteed. Thus, the PR tracking error dynamics is formulated as
(41)ėQ=HQ+GQδe0+DQ−kα2eα−γ¨ref−α¨f.Then, NDO forDQ is constructed as
(42)ṗQ=−lQλQQ+pQ+HQ+GQδe0,lQ=∂λQQ∂Q,D^Q=pQ+λQQ.Defining estimation error asD~Q=DQ−D^Q, it has
(43)D~̇Q=−lQD~Q+ḊQ.∑i=12ψiη¨i/Iyy is bounded according to the corresponding analysis in [25, 26]. Considering the bounded ḋQ, ḋQ,δ, and ∑i=12ψiη¨i/Iyy-differentiator, ḊQ is bounded. Similarly, D~Q is bounded for the fact that ḊQ is bounded, i.e., D~Q≤ΘQ with ΘQ>0.We design the control law to regulate theėQ system as
(44)δe0=1GQ−kQeQ+kαeα2−eα−HQ+γ¨ref+α¨f−D^Q−Θ^QtanhceQεQ,where Θ^Q is governed by the adaptive law
(45)Θ^̇Q=σQ−kQ′Θ^Q−eQtanhceQεQ,with σQ>0, εQ>0, and kQ′>0.


#### 3.2.3. Control Allocation Design for CDA-EDA

In this subsection, a control allocation algorithm on CDA and EDA is integrated to the proposed control considering the hard magnitude and rate constraints of canard and elevator deflections.Considering the componentsM in (10), we briefly define
(46)uMcom=CMδcδccom+CMδeδecom,uMdes=CMδcδc0+CMδeδe0,where uMcom denotes the commands generated by control allocation; uMdes presents the desired values.Meanwhile, assume the future control sequence withz prediction intervals at the current interval k as
(47)Ξk=δccomkkδccomk+1k⋯δccomk+z−1kδecomkkδecomk+1k⋯δecomk+z−1kT.As the flight control system is a sampled-data system, the magnitude and rate can establish the following simple first-order approximation:(48)δ̇≈δt−δt−tsts,where ts is the control interval and δt−ts is the CDA/EDA command generated in the last control interval; the rate constraint can be absorbed into the magnitude constraint:
(49)δt−ts+RLts≤δt≤δt−ts+RUts.And new constraints can be defined as(50)Mmax=maxMU,δt−ts+RUts,Mmin=minML,δt−ts+RLts.Therefore, the optimization problem of the above control allocation can be expressed in the following form:(51)minΞkJΞksubjecttoMmin≤δ≤Mmaxwith
(52)JΞk=JΔuΞk+JΔδΞk,(53)JΔuΞk=∑j=1zwjuuMcomk+j∣k−uMdesk+j∣k2,(54)JΔδΞk=∑j=1zwjcuccomk+j∣k−ucdesk+j−1∣k2+∑j=1zwjeuecomk+j∣k−uedesk+j−1∣k2,where wj∗ is the tuning weight matrices for the jth step.Remark 1.
The cost function in (52) is set for two main practical purposes. One goal reflected in JΔuΞk is to pursue approximating precision, and the other goal reflected in JΔδΞk is to maximize energy savings.


## 3.1. Velocity Tracking Control Design

This section presents an adaptive tracking controller for velocity subsystem (1) ensuring V→Vref. The velocity dynamics is first expressed in the following form:
(19)V̇=HV+GVΦ+DV,where
(20)HV=q¯SmCT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0cosα−CD2α2+CD1α+CD0−gsinγ,GV=CT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0cosα,DV=−q¯SCDδe2δe2+CDδeδe+CDδc2δc2+CDδcδc+dV.First, similar to [23], we introduce a modified tracking error to handle the saturated Φ as
(21)eV=V−Vref−χV,where χV is governed by
(22)χ̇V=−kVχV+GVΦ−ΦV,with kV>0. Then, ėV is calculated as
(23)ėV=kVχV+HV+GVΦ+DV−V̇ref.Taking into account the boundedḋV and the finite deflecting velocity of physical control surfaces, ḊV is assumed to be bounded. Hence, the NDO for DV is constructed.
(24)ṗV=−lVλVV+pV+HV+GVΦc,lV=∂λVV∂V,D^V=pV+λVV.Defining estimation error asD~V=DV−D^V, it has
(25)D~̇V=−lVD~V+ḊV.For the fact thatḊV is bounded, D~V is also bounded from Lemma 3, i.e., D~V≤ΘV with ΘV>0.We design the control law(26)Φc=1GV−kVV−Vref−HV+V̇ref−DV−Θ^VtanhceVεV,where D^V is estimated by NDO and Θ^V is the estimate of ΘV, generated by the adaptive law:
(27)Θ^̇V=σV−kV′Θ^V−eVtanhceVεV,where σV>0, kV′>0, and εV>0.


## 3.2. Altitude Tracking Control Design

In the altitude subsystem, control allocation is employed as an attractive technology to handle both canard and elevator servomechanism dynamics and constraints. For convenience,δc0 and δe0 represent the desired control command to be designed, δccom and δecom denote the control command after the process of control allocation, and δc and δe are the real control inputs applied to AHVs with the dynamics δc/δccom=20/s+20 and δe/δecom=20/s+20 [24]. We also define Δδc,ΔδeT=δc−δc0,δe−δe0T for the following design.


### 3.2.1. Design forh,γ-Dynamics

Recalling the kinematical equation (2), the realization of altitude reference tracking can be transferred to the FPA tracking problem. In this paper, to achieve h→href, the control system steers the AHVs to track the FPA reference:
(28)γref=arcsin−kVh−href+ḣrefV,where href is the smooth altitude reference generated by the traditional second-order command filter; kh>0.Define a time-related function asαf, which decides the final AoA. The FPA dynamics can be calculated as
(29)γ̇=RΓα−αf+Hγ+Gγδc0+Dγ,where
(30)RΓ=q¯SmVCL1+CT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0Φ+CT3αf3+CT2αf2+CT1αf+CT0,Hγ=q¯SmVCL1+CT,Φ3αf3+CT,Φ2αf2+CT,Φ1αf+CT,Φ0Φ+CT3αf3+CT2αf2+CT1αf+CT0αf+q¯SmVCL0−gcosγV,Gγ=q¯SmVCLδc,Dγ=dγ,αf+dγ,δ+dγ,dγ,αf=q¯SmVCT,Φ3α3+CT,Φ2α2+CT,Φ1αΦ+CT3α3+CT2α2+CT1α−CT,Φ3αf3+CT,Φ2αf2+CT,Φ1αfΦ−CT3αf3+CT2αf2+CT1αfsinα,dγ,δ=q¯SmVCLδeδe+CLδcΔδc.Considering the real smallα, the approximate residual dγ,αf is assumed to be bounded. Meanwhile, the unknown disturbance introduced by the above control allocation dγ,δ is bounded considering the amplitude and rate constrained actuators. Thus, the boundedness of the disturbance term Dγ is guaranteed.Furthermore, similar to [4], the virtual AoA control is designed as
(31)αd=αf−eγ,α−αf=α−αd−eγ=eα−eγ.Then,ėγ can be rewritten as
(32)ėγ=RΓeα−eγ+Hγ+Gγδc0+Dγ−γ̇ref.Similarly, the following NDO forDγ is constructed.
(33)ṗγ=−lγλγγ+pγ+RΓeα−eγ+Hγ+Gγδc0,lγ=∂λγγ∂γ,D^γ=pγ+λγγ.Defining estimation error asD~γ=Dγ−D^γ, and
(34)D~̇γ=−lγD~γ+Ḋγ.Further noting the boundeddγ,αf, dγ,δ, and dγ, the boundedness of Dγ is guaranteed. From Lemma 3, D~γ is bounded for the fact that Ḋγ is bounded, i.e., D~γ≤Θγ with Θγ>0.We design the control law to regulate theėγ system as
(35)δc0=1Gγ−kγeγ−RΓeα−Hγ+γ̇ref−Dγ−Θ^γtanhceγεγ,where Θ^γ is governed by the adaptive law
(36)Θ^̇γ=σγ−kγ′Θ^γ+eγtanhceγεγ,with σγ>0, εγ>0, and kγ′>0.


### 3.2.2. Design forα,Q-Dynamics

For the altitude subsystem, the AoA tracking error dynamics have the relationship(37)ėα=α̇−α̇d=α̇−α̇f+ėγ=Q−γ̇ref−α̇f.DefineeQ=Q−Qd with Qd=−kαeα+γ̇ref+α̇f, and ėα can be rewritten as
(38)ėα=−kαeα+eQ,Q̇d=kα2eα−kαeQ+γ¨ref+α¨f.Then, the PR dynamics is formulated as(39)Q̇=HQ+GQδe0+DQ,where
(40)HQ=zTq¯SIyyCT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0Φ+CT3α3+CT2α2+CT1α+CT0,GQ=q¯c¯SIyyCMδe,DQ=dQ+dQ,δ+∑i=12ψiη¨iIyy,dQ,δ=q¯c¯SIyyCMδcδc+CMδeΔδe.Following a similar analysis to previous steps, the boundedness ofDQ is also guaranteed. Thus, the PR tracking error dynamics is formulated as
(41)ėQ=HQ+GQδe0+DQ−kα2eα−γ¨ref−α¨f.Then, NDO forDQ is constructed as
(42)ṗQ=−lQλQQ+pQ+HQ+GQδe0,lQ=∂λQQ∂Q,D^Q=pQ+λQQ.Defining estimation error asD~Q=DQ−D^Q, it has
(43)D~̇Q=−lQD~Q+ḊQ.∑i=12ψiη¨i/Iyy is bounded according to the corresponding analysis in [25, 26]. Considering the bounded ḋQ, ḋQ,δ, and ∑i=12ψiη¨i/Iyy-differentiator, ḊQ is bounded. Similarly, D~Q is bounded for the fact that ḊQ is bounded, i.e., D~Q≤ΘQ with ΘQ>0.We design the control law to regulate theėQ system as
(44)δe0=1GQ−kQeQ+kαeα2−eα−HQ+γ¨ref+α¨f−D^Q−Θ^QtanhceQεQ,where Θ^Q is governed by the adaptive law
(45)Θ^̇Q=σQ−kQ′Θ^Q−eQtanhceQεQ,with σQ>0, εQ>0, and kQ′>0.


### 3.2.3. Control Allocation Design for CDA-EDA

In this subsection, a control allocation algorithm on CDA and EDA is integrated to the proposed control considering the hard magnitude and rate constraints of canard and elevator deflections.Considering the componentsM in (10), we briefly define
(46)uMcom=CMδcδccom+CMδeδecom,uMdes=CMδcδc0+CMδeδe0,where uMcom denotes the commands generated by control allocation; uMdes presents the desired values.Meanwhile, assume the future control sequence withz prediction intervals at the current interval k as
(47)Ξk=δccomkkδccomk+1k⋯δccomk+z−1kδecomkkδecomk+1k⋯δecomk+z−1kT.As the flight control system is a sampled-data system, the magnitude and rate can establish the following simple first-order approximation:(48)δ̇≈δt−δt−tsts,where ts is the control interval and δt−ts is the CDA/EDA command generated in the last control interval; the rate constraint can be absorbed into the magnitude constraint:
(49)δt−ts+RLts≤δt≤δt−ts+RUts.And new constraints can be defined as(50)Mmax=maxMU,δt−ts+RUts,Mmin=minML,δt−ts+RLts.Therefore, the optimization problem of the above control allocation can be expressed in the following form:(51)minΞkJΞksubjecttoMmin≤δ≤Mmaxwith
(52)JΞk=JΔuΞk+JΔδΞk,(53)JΔuΞk=∑j=1zwjuuMcomk+j∣k−uMdesk+j∣k2,(54)JΔδΞk=∑j=1zwjcuccomk+j∣k−ucdesk+j−1∣k2+∑j=1zwjeuecomk+j∣k−uedesk+j−1∣k2,where wj∗ is the tuning weight matrices for the jth step.Remark 1.
The cost function in (52) is set for two main practical purposes. One goal reflected in JΔuΞk is to pursue approximating precision, and the other goal reflected in JΔδΞk is to maximize energy savings.


## 3.2.1. Design forh,γ-Dynamics

Recalling the kinematical equation (2), the realization of altitude reference tracking can be transferred to the FPA tracking problem. In this paper, to achieve h→href, the control system steers the AHVs to track the FPA reference:
(28)γref=arcsin−kVh−href+ḣrefV,where href is the smooth altitude reference generated by the traditional second-order command filter; kh>0.Define a time-related function asαf, which decides the final AoA. The FPA dynamics can be calculated as
(29)γ̇=RΓα−αf+Hγ+Gγδc0+Dγ,where
(30)RΓ=q¯SmVCL1+CT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0Φ+CT3αf3+CT2αf2+CT1αf+CT0,Hγ=q¯SmVCL1+CT,Φ3αf3+CT,Φ2αf2+CT,Φ1αf+CT,Φ0Φ+CT3αf3+CT2αf2+CT1αf+CT0αf+q¯SmVCL0−gcosγV,Gγ=q¯SmVCLδc,Dγ=dγ,αf+dγ,δ+dγ,dγ,αf=q¯SmVCT,Φ3α3+CT,Φ2α2+CT,Φ1αΦ+CT3α3+CT2α2+CT1α−CT,Φ3αf3+CT,Φ2αf2+CT,Φ1αfΦ−CT3αf3+CT2αf2+CT1αfsinα,dγ,δ=q¯SmVCLδeδe+CLδcΔδc.Considering the real smallα, the approximate residual dγ,αf is assumed to be bounded. Meanwhile, the unknown disturbance introduced by the above control allocation dγ,δ is bounded considering the amplitude and rate constrained actuators. Thus, the boundedness of the disturbance term Dγ is guaranteed.Furthermore, similar to [4], the virtual AoA control is designed as
(31)αd=αf−eγ,α−αf=α−αd−eγ=eα−eγ.Then,ėγ can be rewritten as
(32)ėγ=RΓeα−eγ+Hγ+Gγδc0+Dγ−γ̇ref.Similarly, the following NDO forDγ is constructed.
(33)ṗγ=−lγλγγ+pγ+RΓeα−eγ+Hγ+Gγδc0,lγ=∂λγγ∂γ,D^γ=pγ+λγγ.Defining estimation error asD~γ=Dγ−D^γ, and
(34)D~̇γ=−lγD~γ+Ḋγ.Further noting the boundeddγ,αf, dγ,δ, and dγ, the boundedness of Dγ is guaranteed. From Lemma 3, D~γ is bounded for the fact that Ḋγ is bounded, i.e., D~γ≤Θγ with Θγ>0.We design the control law to regulate theėγ system as
(35)δc0=1Gγ−kγeγ−RΓeα−Hγ+γ̇ref−Dγ−Θ^γtanhceγεγ,where Θ^γ is governed by the adaptive law
(36)Θ^̇γ=σγ−kγ′Θ^γ+eγtanhceγεγ,with σγ>0, εγ>0, and kγ′>0.


## 3.2.2. Design forα,Q-Dynamics

For the altitude subsystem, the AoA tracking error dynamics have the relationship(37)ėα=α̇−α̇d=α̇−α̇f+ėγ=Q−γ̇ref−α̇f.DefineeQ=Q−Qd with Qd=−kαeα+γ̇ref+α̇f, and ėα can be rewritten as
(38)ėα=−kαeα+eQ,Q̇d=kα2eα−kαeQ+γ¨ref+α¨f.Then, the PR dynamics is formulated as(39)Q̇=HQ+GQδe0+DQ,where
(40)HQ=zTq¯SIyyCT,Φ3α3+CT,Φ2α2+CT,Φ1α+CT,Φ0Φ+CT3α3+CT2α2+CT1α+CT0,GQ=q¯c¯SIyyCMδe,DQ=dQ+dQ,δ+∑i=12ψiη¨iIyy,dQ,δ=q¯c¯SIyyCMδcδc+CMδeΔδe.Following a similar analysis to previous steps, the boundedness ofDQ is also guaranteed. Thus, the PR tracking error dynamics is formulated as
(41)ėQ=HQ+GQδe0+DQ−kα2eα−γ¨ref−α¨f.Then, NDO forDQ is constructed as
(42)ṗQ=−lQλQQ+pQ+HQ+GQδe0,lQ=∂λQQ∂Q,D^Q=pQ+λQQ.Defining estimation error asD~Q=DQ−D^Q, it has
(43)D~̇Q=−lQD~Q+ḊQ.∑i=12ψiη¨i/Iyy is bounded according to the corresponding analysis in [25, 26]. Considering the bounded ḋQ, ḋQ,δ, and ∑i=12ψiη¨i/Iyy-differentiator, ḊQ is bounded. Similarly, D~Q is bounded for the fact that ḊQ is bounded, i.e., D~Q≤ΘQ with ΘQ>0.We design the control law to regulate theėQ system as
(44)δe0=1GQ−kQeQ+kαeα2−eα−HQ+γ¨ref+α¨f−D^Q−Θ^QtanhceQεQ,where Θ^Q is governed by the adaptive law
(45)Θ^̇Q=σQ−kQ′Θ^Q−eQtanhceQεQ,with σQ>0, εQ>0, and kQ′>0.


## 3.2.3. Control Allocation Design for CDA-EDA

In this subsection, a control allocation algorithm on CDA and EDA is integrated to the proposed control considering the hard magnitude and rate constraints of canard and elevator deflections.Considering the componentsM in (10), we briefly define
(46)uMcom=CMδcδccom+CMδeδecom,uMdes=CMδcδc0+CMδeδe0,where uMcom denotes the commands generated by control allocation; uMdes presents the desired values.Meanwhile, assume the future control sequence withz prediction intervals at the current interval k as
(47)Ξk=δccomkkδccomk+1k⋯δccomk+z−1kδecomkkδecomk+1k⋯δecomk+z−1kT.As the flight control system is a sampled-data system, the magnitude and rate can establish the following simple first-order approximation:(48)δ̇≈δt−δt−tsts,where ts is the control interval and δt−ts is the CDA/EDA command generated in the last control interval; the rate constraint can be absorbed into the magnitude constraint:
(49)δt−ts+RLts≤δt≤δt−ts+RUts.And new constraints can be defined as(50)Mmax=maxMU,δt−ts+RUts,Mmin=minML,δt−ts+RLts.Therefore, the optimization problem of the above control allocation can be expressed in the following form:(51)minΞkJΞksubjecttoMmin≤δ≤Mmaxwith
(52)JΞk=JΔuΞk+JΔδΞk,(53)JΔuΞk=∑j=1zwjuuMcomk+j∣k−uMdesk+j∣k2,(54)JΔδΞk=∑j=1zwjcuccomk+j∣k−ucdesk+j−1∣k2+∑j=1zwjeuecomk+j∣k−uedesk+j−1∣k2,where wj∗ is the tuning weight matrices for the jth step.Remark 1.
The cost function in (52) is set for two main practical purposes. One goal reflected in JΔuΞk is to pursue approximating precision, and the other goal reflected in JΔδΞk is to maximize energy savings.


## 4. Stability Analysis

In this section, stability and performance properties of the closed-loop systems are summarised.Consider the Lyapunov functionL=LV+Lh,γ+Lα,Q, where
(55)LV=12eV2+1σVΘ~V2,Lh,γ=12eγ2+1σγΘ~γ2,Lα,Q=12eα2+eQ2+1σQΘ~Q2.Utilizing (23), (27) together with Lemma 1 and applying inequalities: 2Θ~VΘ^V≤ΘV2−Θ~V2, L̇V satisfies
(56)L̇V=eVėV+1σVΘ~VΘ~̇V≤−kVeV2−kV′2Θ~V2+kV′2ΘV+εVΘV.Similarly,Lh,γ and Lα,Q can be calculated from (32), (35), (36) and (41), (44), (45), respectively.
(57)L̇h,γ=eγėγ+1σγΘ~γΘ~̇γ≤−kγ+RΓeγ2−kγ′2Θ~γ2+kγ′2Θγ+εγΘγ,(58)L̇α,Q=eαėα+eQėQ+1σQΘ~QΘ~̇Q≤−kαeα2−kQ−kαeQ2−kQ′2Θ~Q2+kQ′2ΘQ+εQΘQ.By adding inequalities (56)–(58), we obtain
(59)L̇≤−kVeV2−kγ+RΓeγ2−kαeα2−kQ−kαeQ2−kV′2Θ~V2−kγ′2Θ~γ2−kQ′2Θ~Q2+kV′2ΘV+εVΘV+kγ′2Θγ+εγΘγ+kQ′2ΘQ+εQΘQ≤−aLL+bL,where kQ>kα, kγ>−RΓ, and aL,bL∈ℝ+ are constants expressed as
(60)aL=min2kV,2kγ+RΓ,2kα,2kQ−kα,kV′σV,kγ′σγ,kQ′σQ,bL=kV′2ΘV+εVΘV+kγ′2Θγ+εγΘγ+kV′2ΘQ+εQΘQ.Obviously,L̇<0 provided that L>bL/aL. Therefore, all signals involved in the Lyapunov function and utilized in the design are uniformly ultimately bounded.Applying Lemma2 to (59), it is concluded that the boundedness of the related variables and signals in the proposed control can be ensured. Meanwhile, the tracking error ex with x=V, h retains the uniformly ultimate boundedness property:
(61)maxeVt,eht≤2bLaL+L0−bLaLe−aLt,with
(62)limt→∞maxeVt,eht≤2bLaL.


## 5. Simulation Results

This section implements a simulation study to verify the proposed controller. In the simulating model, the nominal parameter values of the AHV model are selected according to [1]. For comparison, a slightly modified version of the adaptive fault-tolerant control presented in [9] is applied to the same scenario (denoted as “compared control” for convenience).The simulated AHV is initially cruising atV0=7846 ft/s, h0=85000 ft, γ0=0 rad, α0=2.6 deg, η10=0.92ft⋅slugs/ft, and η20=0.61ft⋅slugs/ft and plans to implement a maneuver in the next 600 s. Φcooling=0.01 and Φchoking is selected according to [12]; ML=−20 deg, MU=20 deg, RL=−60 deg/s, and RU=60 deg/s; the simulating control interval is set as ts=10 ms.Velocity and altitude references are generated by passing step command signals:(63)Vct=7,846,t=0s,9,846,t∈0s,300s,9,000,t∈300s,600s,hct=85,000,t=0s,110,000,t∈0s,300s,95,000,t∈300s,600s,through a second-order filter [4] with the damping ratio of 0.9 and the natural frequency of 0.03 rad/s.The control parameters are set as follows. In the velocity loop,kV=5, kV′=10, σV=0.1, εV=1, and lV=5. In the altitude loop, kγ=20, kγ′=1, σγ=0.5, εγ=5, lγ=10; kα=10, kQ=40, kQ′=0.3, σQ=0.8, εQ=10, and lQ=20.The main simulating results are given in Figures1–9. Figures 1 and 2 show the output tracking results during the ascending maneuver and the descending maneuver, from which we see that the proposed control performs better, though both controls achieve the desired conditions. Time histories of flight attitudes including FPA, AoA, and PR are given in Figure 3. It is seen from Figure 1 that the desired velocity reference cannot be tracked because of the FER saturation property, also see Figure 5. In this situation, the employed auxiliary system generates a modified reference, which is less aggressive and can be well tracked within the ability of scramjet. Figure 4 shows the flexible states in the comparison simulation. Figures 5–7 indicate the control input signals of FER, CDA, and EDA. It is seen from Figures 6 and 7 that both the magnitude constraints and rate constraints of CDA/EDA are well constrained within prescribed bound under the proposed control, which performs better than the compared control. Figure 8 shows the results of lumped disturbance estimation from NDOs. Θ^V, Θ^γ, and Θ^Q estimated by the adaptive laws are shown in Figure 9.Figure 1
Velocity tracking results in the comparison simulation: (a) velocity tracking; (b) velocity tracking error.
(a)(b)Figure 2
Altitude tracking results in the comparison simulation: (a) altitude tracking; (b) altitude tracking error.
(a)(b)Figure 3
Flight attitudes in the comparison simulation: (a) flight path angle; (b) angle of attack; (c) pitch rate.
(a)(b)(c)Figure 4
Flexible states in the comparison simulation.Figure 5
FER and state of the auxiliary systemχV: (a) FER; (b) state of the auxiliary system.
(a)(b)Figure 6
CDA and CDA rate in the comparison simulation: (a) canard deflection angle; (b) canard deflection angle rate.
(a)(b)Figure 7
EDA and EDA rate in the comparison simulation: (a) elevator deflection angle; (b) elevator deflection angle rate.
(a)(b)Figure 8
Results of lumped disturbance estimation.Figure 9
Estimation errorsΘ^V, Θ^γ, and Θ^Q.


## 6. Conclusions

In this paper, a disturbance observer-based adaptive control for a hypersonic vehicle has been addressed to obtain the capability of handling external disturbances and actuator constraints. Fast disturbance estimate has been addressed via NDO and adaptive control compensation in each subsystem. An auxiliary system has been employed subject to the scramjet input saturation. Meanwhile, there is an effective mechanism of dealing with the magnitude/rate constraints and dynamics of aerodynamic control surfaces by introducing a control allocation module. Simulations have been provided to verify the effectiveness of the proposed control.



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*Source: 8856676-2020-10-10.xml*