# Robust Adaptive Attitude Control for Airbreathing Hypersonic Vehicle with Attitude Constraints and Propulsive Disturbance

Authors: Jian Fu; Liangming Wang; Mou Chen; Sijiang Chang
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

A robust adaptive backstepping attitude control scheme, combined with invariant-set-based sliding mode control and fast-nonlinear disturbance observer, is proposed for the airbreathing hypersonic vehicle with attitude constraints and propulsive disturbance. Based on the positive invariant set and backstepping method, an innovative sliding surface is firstly developed for the attitude constraints. And the propulsive disturbance of airbreathing hypersonic vehicle is described as a differential equation which is motivated by attitude angles in this paper. Then, an adaptive fast-nonlinear disturbance observer for the proposed sliding surface is designed to estimate this kind of disturbance. The convergence of all closed-loop signals is rigorously proved via Lyapunov analysis method under the developed robust attitude control scheme. Finally, simulation results are given to illustrate the effectiveness of the proposed attitude control scheme.


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## Body

## 1. Introduction

In recent years, a new kind of aerospace vehicle, which is called airbreathing hypersonic vehicle (AHV), has attracted the considerable attention in both military and civil communities. Comparing with conventional flight vehicles, this new aerospace vehicle can sustain flight for a long time with high flight speed and cover a large envelope [1, 2]. Due to the strong disturbance and fast changed pneumatics parameters, it is still a challenging task to design a robust AHV control system. Therefore, considerable efforts have been made to develop an efficient flight controller for this kind of aerospace vehicles. In [3, 4], adaptive function link network control and fuzzy logical system were proposed for the aerospace vehicle with uncertainties. And baseline nonlinear model predictive controller was presented in [5] for AHV with nonvanishing mismatched disturbances. In [6], an adaptive fault-tolerant control strategy with input saturation is proposed for AHV flight system with actuator faults and external disturbance. However, the robust attitude control method should be further developed for the AHV with attitude constraints and propulsive disturbance.In order to suppress the unknown external disturbance from changeable flight environment, many papers have focused on the disturbance observer of AHV flight control system [7–11]. A nonlinear disturbance observer based control (NOBC) method is proposed in [7] for the longitudinal dynamics of a generic airbreathing hypersonic vehicle under mismatched disturbance. In [8], the NDO technique is combined with Takagi-Sugeno (T-S) fuzzy linear model to design a disturbance observer based guaranteed cost fuzzy controller (GCFC) for the AHV flight systems. To achieve satisfactory tracking performance, a composite tracking controller is proposed for AHV using NDO and backstepping technologies [9]. However, there are few disturbance observer based control results for AHV with the propulsive disturbance. This kind of disturbance for AHV is usually motivated from the coupling between the scramjet propulsion system and attitude angles [10, 11]. Thus, this kind of disturbance will lead to a challenging problem for the attitude control of the AHV. In this paper, the propulsive disturbance is described as a differential equation which is motivated by attitude constraints in this paper. And a composite controller is proposed to handle this disturbance using adaptive NDO and backstepping technologies.Furthermore, sliding mode control (SMC) is considered as an important method due to its high precision control, relative simplicity of control design, and high robust features with respect to system internal perturbations and external disturbances. Thus, the SMC and its application of AHV have been widely studied in [12–21]. A composite control treatment based on sliding model method is presented to the nonlinear longitudinal dynamics of AHV vehicles [12]. And dynamic terminal sliding mode technique is proposed for the robust control design of AHV [13, 14]. Considering the input constraints, a second-order terminal sliding mode control method is discussed for AHV flight system in reentry phase [15]. Meanwhile, an adaptive dynamic sliding mode control method is presented for the fault-tolerant control problem of AHV in [16]. However, most of the AHV control systems combining backstepping control with SMC have not considered the attitude constraints. Thus, it is necessary to design an innovative sliding mode control for the AHV using backstepping technique and considering the attitude constraints.This work is motivated by the robust attitude control of AHV with attitude constraints and propulsive disturbance. The control objective is that the proposed robust control scheme can track a desired trajectory in the presence of the unknown propulsive disturbance and attitude constraints. An innovative sliding surface is firstly developed in this paper. And the invariant sets and saturation function are utilized in this surface for the attitude constraints of AHV. To guarantee the control effects under propulsive disturbance, the switching information from invariant-set-based sliding surface is included in the adaptive disturbance observer to increase the convergence speed. Therefore, the organization of the paper is as follows. Section2 details the problem formulation, while main results are given in Section 3. Simulation results are presented in Section 4 to show the effectiveness of proposed robust adaptive backstepping attitude control for AHV. And some conclusion results are shown in Section 5.


## 2. Problem Statement

### 2.1. Model Description

To study the robust attitude controller, a nonlinear attitude motion model of AHV flight system is described as the following nonlinear affine nonlinear system [22–24]:(1)x˙1=Fsx1+Gsx1x2,x˙2=Ffx1,x2+Gfx1,x2u+dt,y=x1,where x1=[α,β,μ]T∈R3 is the attitude angle vector of slow-loop states, x2=[p,q,r]T∈R3 is the body-axis angular rate vector of fast-loop states, u=[δe,δa,δr,δx,δy,δz]T∈R6 is the deflection vector of control surfaces which are the system control inputs, y∈R3 is the system output, and the unknown disturbance d(y,yd,t)∈R3 is defined as(2)dy,yd,t=d1y,yd,t+d2t,where d1(y,yd,t)∈R3 is the unknown propulsive disturbance which is coupling with the attitude angles and d2(t)∈R3 is the rest of unknown disturbance. The detailed expression of Fs∈R3,  Gs∈R3×3,  Ff∈R3, and Gf∈R3×6 can be found in [24].


### 2.2. Assumptions and Control Objective

In order to reduce the coupling effects between the scramjet propulsion system and attitude angles, the attitude constraints are defined as(3)Ψ=y∈R3∣ydi-ci≤yi≤ydi+ci,ci∈R+,i=1,2,3,where y=[y1,y2,y3]T is the attitude angle vector and yd=[yd1,yd2,yd(3)]T is the desired output signal. When the attitude angle vector y satisfies constraints (3), the propulsive disturbance d1(y,t) will be a small value which can be neglected in the control design process. However, if y exceeds the boundary of constraints (3), this disturbance will be increased significantly. Then, it is supposed that the disturbances d1(y,t) and d2(t) will satisfy the following assumptions.Assumption 1.
For allt>0, there exists Δi such that yd(i)(t)≤Δi,  i=1,2,3.Assumption 2.
The propulsive disturbanced1(y,yd,t) can be described as the following differential equation:(4)d˙1y,yd,t=τ1y,t·δ1εy,yd,where τ1y,t=diag⁡{τ11,τ12,τ13}, and we can obtain τ1(y,t)=[0,0,0]T if y∈ψ. For the element τ1i, there exists an unknown positive constant κi such that τ1i≤κi,  i=1,2,3. The vector δ1ε∈R3 is a special designed sliding surface as shown in (22), and there exists y∈Ψ while δ1ε=0.Assumption 3.
The disturbanced2(t) can be described as the following differential equation:(5)d˙2t=τ2t,where τ2(t)=[τ21,τ22,τ23]T, and there exists a known positive constant μi such that τ2i(t)≤μi,  i=1,2,3.Assumption 4.
The generalized matrix inverses ofGs and Gf are always existing for the nonlinear attitude motion model of the AHV, and the elements in matrixes Fs, Gs, Ff, and Gf are always continuous for the nonlinear attitude motion model of the AHV.Definition 1 (see [25]).
The setQ is said to be positively invariant (PI) for nonlinear system (1), if, for all x1(t1)∈Q, there exists the solution x1(t2)∈Q,  ∀t2>t1.


## 2.1. Model Description

To study the robust attitude controller, a nonlinear attitude motion model of AHV flight system is described as the following nonlinear affine nonlinear system [22–24]:(1)x˙1=Fsx1+Gsx1x2,x˙2=Ffx1,x2+Gfx1,x2u+dt,y=x1,where x1=[α,β,μ]T∈R3 is the attitude angle vector of slow-loop states, x2=[p,q,r]T∈R3 is the body-axis angular rate vector of fast-loop states, u=[δe,δa,δr,δx,δy,δz]T∈R6 is the deflection vector of control surfaces which are the system control inputs, y∈R3 is the system output, and the unknown disturbance d(y,yd,t)∈R3 is defined as(2)dy,yd,t=d1y,yd,t+d2t,where d1(y,yd,t)∈R3 is the unknown propulsive disturbance which is coupling with the attitude angles and d2(t)∈R3 is the rest of unknown disturbance. The detailed expression of Fs∈R3,  Gs∈R3×3,  Ff∈R3, and Gf∈R3×6 can be found in [24].


## 2.2. Assumptions and Control Objective

In order to reduce the coupling effects between the scramjet propulsion system and attitude angles, the attitude constraints are defined as(3)Ψ=y∈R3∣ydi-ci≤yi≤ydi+ci,ci∈R+,i=1,2,3,where y=[y1,y2,y3]T is the attitude angle vector and yd=[yd1,yd2,yd(3)]T is the desired output signal. When the attitude angle vector y satisfies constraints (3), the propulsive disturbance d1(y,t) will be a small value which can be neglected in the control design process. However, if y exceeds the boundary of constraints (3), this disturbance will be increased significantly. Then, it is supposed that the disturbances d1(y,t) and d2(t) will satisfy the following assumptions.Assumption 1.
For allt>0, there exists Δi such that yd(i)(t)≤Δi,  i=1,2,3.Assumption 2.
The propulsive disturbanced1(y,yd,t) can be described as the following differential equation:(4)d˙1y,yd,t=τ1y,t·δ1εy,yd,where τ1y,t=diag⁡{τ11,τ12,τ13}, and we can obtain τ1(y,t)=[0,0,0]T if y∈ψ. For the element τ1i, there exists an unknown positive constant κi such that τ1i≤κi,  i=1,2,3. The vector δ1ε∈R3 is a special designed sliding surface as shown in (22), and there exists y∈Ψ while δ1ε=0.Assumption 3.
The disturbanced2(t) can be described as the following differential equation:(5)d˙2t=τ2t,where τ2(t)=[τ21,τ22,τ23]T, and there exists a known positive constant μi such that τ2i(t)≤μi,  i=1,2,3.Assumption 4.
The generalized matrix inverses ofGs and Gf are always existing for the nonlinear attitude motion model of the AHV, and the elements in matrixes Fs, Gs, Ff, and Gf are always continuous for the nonlinear attitude motion model of the AHV.Definition 1 (see [25]).
The setQ is said to be positively invariant (PI) for nonlinear system (1), if, for all x1(t1)∈Q, there exists the solution x1(t2)∈Q,  ∀t2>t1.


## 3. Robust Adaptive Backstepping Attitude Control for the AHV with Attitude Constraints and Propulsive Disturbance

### 3.1. Preliminary

In this section, a few results are given to support the design process of robust adaptive backstepping attitude controller. Considering the standard backstepping control design method, we define(6)e1=y-yd,e2=x2-ν1,where ν1∈R3 is a designed virtual control law. Then, considering (1), the derivatives e˙1,e˙2 can be written as(7)e˙1=Fsx1+Gsx1·e2+ν1-y˙d,(8)e˙2=Ffx1,x2+Gfx1,x2·u+dy,yd,t-ν˙1,where e1=[e11,e12,e13]T∈R3 are the state errors of slow-loop states and e2=[e21,e22,e23]T∈R3 are the state errors of fast-loop states. Invoking (6), the attitude constraints (3) can be rewritten as(9)Ψ=e1,∫e1∣-ci≤e1i≤ci,ci∈R+,where Ψ=[Ψ1,Ψ2,Ψ3]T and Ψi=e1i,∫e1i∣-ci≤e1i≤ci.Considering the attitude constraints of AHV, the unidirectional auxiliary surfaces (UAS) for slow-loop and fast-loop states are utilized to design the invariant-set-based sliding mode controller. The detailed design process for these surfaces is given as follows.Step 1.
Considering the slow-loop system (7), we define the switching surfaces as(10)S11=e1+ξ11·∫e1=0,S12=e1+ξ12·∫e1=0,where ξ11=diag⁡{ξ111,ξ112,ξ113},  ξ12=diag⁡{ξ121,ξ122,ξ123},  S11=[S111,S112,S113]T, S12=[S121,S122,S123]T,  ξ11i>ξ12i>0,  i=1,2,3, and ∫e1dt is denoted by ∫e1 for brevity. The conditions ξ11i>0 and  ξ12i>0 are given to guarantee the stability of switching surfaces S11i=0 and S12i=0. And the condition ξ11i>ξ12i is used to avoid the overlap of switching surfaces.Step 2.
Invoking the coefficientsξ11i,ξ12i in (10), the unidirectional auxiliary surface h1i can be designed as(11)h1i=ω1i1·e1i+ω1i2·∫e1i+M1i,where (12)ω1i1=1S11i<0;S12i<0ω11i1S11i<0;S12i≥0ω12i1S11i≥0;S12i<0-1S11i≥0;S12i≥0;ω1i2=ω10i2S11i<0;S12i<0ω11i2S11i<0;S12i≥0ω12i2S11i≥0;S12i<0-ω13i2S11i≥0;S12i≥0, where ω10i2,ω13i2 are the designed parameters which satisfy ξ12i<ω10i2<ξ11i,  ξ12i<ω13i2<ξ11i, and  M1i>0, and the rest of coefficients are given as follows:(13)ω11i1=-ξ11i+ξ12i+ω10i2+ω13i2ξ11i-ξ12i,ω11i2=-2ξ11iξ12i+ω10i2ξ11i+ω13i2ξ12iξ11i-ξ12i,ω12i1=ξ11i+ξ12i-ω10i2+ω13i2ξ11i-ξ12i,ω12i2=2ξ11iξ12i-ω13i2ξ11i+ω10i2ξ12iξ11i-ξ12i,where ω11i1≠0 and  ω12i1≠0. Then, the compact form of unidirectional auxiliary surface(14)h1=Ω11·e1+Ω12·∫e1+M1,where h1=[h11,h12,h13]T,  ∫e1=[∫e11,∫e12,∫e13]T,  Ω11=diag⁡{ω111,ω121,ω131},  Ω12=diag⁡{ω112,ω122,ω132},   and M1=[M11,M12,M13]T is a constant vector.
As shown in Figure1, the UAS can form a convex set Qi which satisfies Qi⊆Ψi, and the expression of set Qi is written as(15)Qi=e1i,∫e1i∣h1i≥0.The compact form of convex set Qi can be written as Q=[Q1,Q2,Q3]T, where Q⊆Ψ implies Qi⊆Ψi,   i=1,2,3.Figure 1
The convex setQi in attitude constraint Ψi.Step 3.
Similarly, we can define the following switching surfaces for the fast-loop system (8):(16)S21=e2+ξ21·∫e2=0,S22=e2+ξ22·∫e2=0,where ξ21=diag⁡{ξ211,ξ212,ξ213}, ξ22=diag⁡{ξ221,ξ222,ξ223},  S21=[S211,S212,S213]T, S22=[S221,S222,S223]T, and  ξ21i>ξ22i>0,   i=1,2,3.Step 4.
Invoking the coefficientsξ21i,ξ22i in (16), the unidirectional auxiliary surface h2i can be designed as(17)h2i=ω2i1·e2i+ω2i2·∫e2i+M2i,where (18)ω2i1=1S21i<0;S22i<0ω21i1S21i<0;S22i≥0ω22i1S21i≥0;S22i<0-1S21i≥0;S22i≥0;ω2i2=ω20i2S21i<0;S22i<0ω21i2S21i<0;S22i≥0ω22i2S21i≥0;S22i<0-ω23i2S21i≥0;S22i≥0,where ξ22i<ω20i2<ξ21i,  ξ22i<ω23i2<ξ21i, and M2i>0 are the designed parameters, and the rest of coefficients can be designed as follows:(19)ω21i1=-ξ21i+ξ22i+ω20i2+ω23i2ξ21i-ξ22i,ω21i2=-2ξ21iξ22i+ω20i2ξ21i+ω23i2ξ22iξ21i-ξ22i,ω22i1=ξ21i+ξ22i-ω20i2+ω23i2ξ21i-ξ22i,ω21i2=2ξ21iξ22i-ω20i2ξ21i+ω23i2ξ22iξ21i-ξ22i,where ω21i1≠0 and   ω22i1≠0. Then, the compact form of unidirectional auxiliary surface(20)h2=Ω21·e2+Ω22·∫e2+M2,where h2=[h21,h22,h23]T,  ∫e2=[∫e21,∫e22,∫e23]T,  Ω21=diag⁡{ω211,ω221,ω231},  Ω22=diag⁡{ω212,ω222,ω232}, and M2=[M21,M22,M23]T is a constant vector. Then, there exist the following results for the unidirectional auxiliary surfaces h1=[h11,h12,h13]T and  h2=[h21,h22,h23]T.Lemma 2.
For the pointsP1(t)=(e1i,∫e1i) and  P2(t)=(e2i,∫e2i),  i=1,2,3, it follows that δ1i(t)=M1i-h1i(P1)≥0 and δ2i(t)=M2i-h2i(P2)≥0. And if δ1i(t)=0, there exists P1=(0,0); if δ2i(t)=0, there exists P2=(0,0).Proof.
See AppendixA.Lemma 3.
For alle1i,e2i∈R,  i=1,2,3, the functions δ1i(t) and δ2i(t) in Lemma 2 are continuous functions.Proof.
See AppendixB.


### 3.2. Robust Adaptive Backstepping Attitude Control Based on Disturbance Observer and Attitude Constraints

In the attitude control design, we combine the backstepping method, adaptive invariant-set-based sliding mode control, and the disturbance observer technique to design a robust adaptive backstepping attitude controller for the AHV system (1) with attitude constraints and propulsive disturbance, and the detailed design process is appended as follows.


#### 3.2.1. Design Process for the Controller

Step 1.
Invoking (14) and (20), we can define δ1 and δ2 as(21)δ1=M1-h1,δ2=M2-h2,where δ1=[δ11,δ12,δ13]T and δ2=[δ21,δ22,δ23]T. Then, the innovative sliding mode surface δ1ε can be defined as(22)δ1ε=δ1-ε1·Satδ1ε1,where δ1ε=[δ1ε1,δ1ε2,δ1ε3]T, ε1=diag⁡{M11,M12,M13}, Sat(δ1/ε1)=[Sat(δ11/M11),Sat(δ12/M12),Sat(δ13/M13)]T, and Sat(·) is the saturation function. From (22), it is clear that(23)δ1εi=δ1i-M1i·Satδ1iM1i=0δ1i≤M1iδ1i-M1iδ1i>M1i.Remark 4.
It is clear thatδ1εi=0 when δ1i=M1i-h1i≤Mi. Considering Qi=e1i,∫e1i∣h1i≥0⊆Ψi in (15), we can obtain that y∈Ψ while δ1ε=0. Therefore, the attitude constraints will be satisfied when system states are sliding on the sliding surface δ1ε=0.Step 2.
In this paper, the nonlinear disturbance observer which is employed to estimated(y,yd,t) is defined as(24)d^=Le2-η,η^˙=Ffx1,x2+Gfx1,x2·u+d^-ν˙1+L-1·Ω21·E,where L=LT>0 is a design parameter which should be chosen to render L-0.5I3×3>0,  E=[1,1,1]T.Assumption 5.
For the estimation errord~=d^-d=[d~1,d~2,d~3]T, there exists positive constant θi such that θi>-τ1i·d~i,  i=1,2,3, where the coefficient τ1i is a bounded element in (4).Step 3.
To suppress the estimation errord~ of disturbance, an adaptive item θ^ is defined as(25)θ^˙=λ·δ1ε,where θ^=[θ^1,θ^2,θ^3]T and λ>0 is a design parameter.Step 4.
Definingθ~=θ^-θ and Δ1ε=diag⁡{δ1ε1,δ1ε2,δ1ε3} and considering (7), the virtual control law ν1 is designed as(26)ν1=Gs-1-Fs+y˙d+Ω11-1-Ω12·e1+N1+Δ1ε·θ^,where N1=[N11,N12,N13]T,  N1i≥0,  i=1,2,3 are designed approaching laws. Using the output of the designed nonlinear disturbance observer (24) and considering Assumptions 3 and 4, the robust attitude control law is designed as(27)u=Gf-1-Ff-d^+ν˙1+Ω21-1-Ω22·e2+N2-Ω21-1Ω11Gse2,where Gf-1 is the generalized matrix inverse of Gf, N2=[N21,N22,N23]T, and  N2i>0.5μi2,  i=1,2,3.Remark 6.
DefiningE=[1,1,1]T and considering Δ1ε=diag⁡{δ1ε1,δ1ε2,δ1ε3} and δ1ε=[δ1ε1,δ1ε2,δ1ε3]T, we can obtain (28)ET·Δ1ε=δ1εT;ET·δ1ε=δ1ε1+δ1ε2+δ1ε3.


#### 3.2.2. Proof for the Stability of Sliding Mode Surface

The discussion for the stability of sliding mode surfaceδ1ε=0 is given in this part. Considering δ1ε=0, the control law in (26) can be rewritten as follows:(29)ν1=Gs-1-Fs+y˙d+Ω11-1-Ω12·e1+N1.Substituting (29) into (7) yields(30)e˙1=Ω11-1-Ω12·e1+N1+Gs·e2.Choose the Lyapunov function candidate as(31)V1=ETδ1.From Lemmas 2 and 3, it is clear that the Lyapunov function V1 is a positive and continuous function. And there exists e1=0 while V1=0. Invoking (14) and (21), the time derivative of V1 can be written as(32)V˙1=ETδ˙1=ET-Ω11e˙1-Ω12e1.Substituting (30) into (32) yields(33)V˙1=-ETN1-ETΩ11Gse2.It is apparent that the first term on the right-hand side of (33) is stable. Furthermore, the second term will be canceled as follows.Invoking (4), it is noted that the time derivative of propulsive disturbance d˙1=0 while δ1ε=0. Then, we can obtain d˙=d˙2=τ2 from (2), (3), and (4) and δ1ε=0. Considering the definition of d~ and substituting (27) into (8) yield(34)e˙2=-d~+Ω21-1-Ω22e2+N2-Ω21-1Ω11Gse2.For considering the stability of all signals for the closed-loop control system, the Lyapunov function candidate is chosen as(35)V=V1+0.5d~Td~+ETδ2.The time derivative of V is(36)V˙=V˙1+d~Td~˙+ETδ˙2.Invoking (8), (21), (24), (27), and Assumption 3, the time derivative is(37)V˙=-ETN1-ETN2-d~TLd~-d~d˙≤-ETN1-ETN2-d~TL-0.5I3×3d~+0.5d˙Td˙<-ETN1-ETN2+0.5μTμ-d~TL-0.5I3×3d~,where μ=[μ1,μ2,μ3]T.Theorem 7.
Considering the error systems (7) and (8) with the unknown propulsive disturbance, the disturbance observer is designed as (24) and the robust backstepping attitude control scheme is proposed as (26) and (27). Then, the attitude control error of AHV is convergent when the system states are sliding on the innovative sliding surface δ1ε=0 in (22).Proof.
Invoking (37) and considering L-0.5I3×3>0,  N1i≥0, and N2i>0.5μi2,  i=1,2,3, we have (38)V˙<-ETN1-ETN2+0.5μTμ-d~TL-0.5I3×3d~=-d~TL-0.5I3×3d~-∑N1i+N2i-0.5μi2<0. Thus, the tracking errors e1,e2 and the disturbance approximation error d~ are convergent while the system states are sliding on the innovative sliding mode surface (22).


#### 3.2.3. Proof for the Reaching Phase for Sliding Mode Surface

The discussion for the reaching phase for sliding surface is given in this part. Choose the Lyapunov function candidate as(39)V1=ETδ1ε+12λθ~Tθ~.From Lemmas 2 and 3, it is clear that the Lyapunov function V1 is a positive and continuous function, and there exist δ1ε=0 while V1=0. Assume that δ1ε≠0 and considering (23), there exists the following equation:(40)δ˙1ε=δ˙1=-Ω11e˙1-Ω12e1.Substituting (7) and (26) into (40) yields(41)δ˙1ε=-Ω11Gse2-N1-Δ1ε·θ^.Invoking (26) and (39), the time derivative of V1 can be written as(42)V˙1=ETδ˙1ε+θTθ^˙λSubstituting (25), (28), and (40) into (42) yields(43)V˙1=ET-Ω11Gse2-N1-Δ1ε·θ^+δ1εTθ~=ET-Ω11Gse2-N1-δ1εT·θ^+δ1εTθ~.Invoking (2), (4), and (5) yields(44)d˙=d˙1+d˙2=τ1·δ1ε+τ2.Considering the stability of all signals for the closed-loop control system, the Lyapunov function candidate is chosen as(45)V=V1+0.5d~Td~+ETδ2.The time derivative of V is(46)V˙=V˙1+d~Td~˙+ETδ˙2=V˙1+d~Td^˙-d˙+ETδ˙2.Invoking (21), (24), (27), (43), and (44), the time derivative of V is(47)V˙=-ETN1+N2-δ1εTθ^+δ1εTθ~+d~T-Ld~-d~Tτ1·δ1ε+τ2≤-ETN1+N2-δ1εTθ^+δ1εTθ~-d~TL-0.5I3×3d~-d~Tτ1δ1ε+0.5τ2Tτ2.Considering Assumption 5, δ1ε>0, (22), and (23), we have(48)V˙≤-ETN1+N2+δ1εT-θ^+θ~-τ1d~-d~TL-0.5I3×3d~+0.5τ2Tτ2≤-ETN1+N2+δ1εT-θ^+θ~+θ-d~TL-0.5I3×3d~+0.5τ2Tτ2.Substituting θ~=θ^-θ into (48) yields(49)V˙≤-ETN1+N2-d~TL-0.5I3×3d~+0.5τ2Tτ2.Theorem 8.
Considering the attitude motion dynamics (1) of the AHV with attitude constraints and unknown propulsive disturbance, the disturbance observer is designed as (24) and the robust backstepping attitude control scheme is proposed as (26) and (27). Then, attitude controller error of AHV is convergent under the attitude constraints (3). Meanwhile, the system states will arrive at the sliding surface δ1ε=0, and this surface also can be proved as a positively invariant set.Proof.
According to (49), we have(50)V˙≤-d~TL-0.5I3×3d~-∑N1i+N2i-0.5μi2.Substituting N1i≥0,  N2i>0.5μi2, and L-0.5I3×3>0 into (50) yields(51)V˙<0.Thus, the system states will arrive at the sliding surface δ1ε=0. From Remark 4, it is noted that attitude constraints (3) will be satisfied when system states are sliding on sliding surface δ1ε=0.
Considering the stability of sliding surface as shown in Theorem8, the attitude control error of AHV is convergent under the attitude constraints (3). From (39), (45), and (51), there exists δ1ε(t)=0 for all t>t1 if V(t1)=0. And the sliding surface δ1ε(t)=0 can be rewritten as a closed set Q as shown in (15) and Remark 4. Then, the sliding surface δ1ε=0 can be seen as a positively invariant set according to Definition 1.


## 3.1. Preliminary

In this section, a few results are given to support the design process of robust adaptive backstepping attitude controller. Considering the standard backstepping control design method, we define(6)e1=y-yd,e2=x2-ν1,where ν1∈R3 is a designed virtual control law. Then, considering (1), the derivatives e˙1,e˙2 can be written as(7)e˙1=Fsx1+Gsx1·e2+ν1-y˙d,(8)e˙2=Ffx1,x2+Gfx1,x2·u+dy,yd,t-ν˙1,where e1=[e11,e12,e13]T∈R3 are the state errors of slow-loop states and e2=[e21,e22,e23]T∈R3 are the state errors of fast-loop states. Invoking (6), the attitude constraints (3) can be rewritten as(9)Ψ=e1,∫e1∣-ci≤e1i≤ci,ci∈R+,where Ψ=[Ψ1,Ψ2,Ψ3]T and Ψi=e1i,∫e1i∣-ci≤e1i≤ci.Considering the attitude constraints of AHV, the unidirectional auxiliary surfaces (UAS) for slow-loop and fast-loop states are utilized to design the invariant-set-based sliding mode controller. The detailed design process for these surfaces is given as follows.Step 1.
Considering the slow-loop system (7), we define the switching surfaces as(10)S11=e1+ξ11·∫e1=0,S12=e1+ξ12·∫e1=0,where ξ11=diag⁡{ξ111,ξ112,ξ113},  ξ12=diag⁡{ξ121,ξ122,ξ123},  S11=[S111,S112,S113]T, S12=[S121,S122,S123]T,  ξ11i>ξ12i>0,  i=1,2,3, and ∫e1dt is denoted by ∫e1 for brevity. The conditions ξ11i>0 and  ξ12i>0 are given to guarantee the stability of switching surfaces S11i=0 and S12i=0. And the condition ξ11i>ξ12i is used to avoid the overlap of switching surfaces.Step 2.
Invoking the coefficientsξ11i,ξ12i in (10), the unidirectional auxiliary surface h1i can be designed as(11)h1i=ω1i1·e1i+ω1i2·∫e1i+M1i,where (12)ω1i1=1S11i<0;S12i<0ω11i1S11i<0;S12i≥0ω12i1S11i≥0;S12i<0-1S11i≥0;S12i≥0;ω1i2=ω10i2S11i<0;S12i<0ω11i2S11i<0;S12i≥0ω12i2S11i≥0;S12i<0-ω13i2S11i≥0;S12i≥0, where ω10i2,ω13i2 are the designed parameters which satisfy ξ12i<ω10i2<ξ11i,  ξ12i<ω13i2<ξ11i, and  M1i>0, and the rest of coefficients are given as follows:(13)ω11i1=-ξ11i+ξ12i+ω10i2+ω13i2ξ11i-ξ12i,ω11i2=-2ξ11iξ12i+ω10i2ξ11i+ω13i2ξ12iξ11i-ξ12i,ω12i1=ξ11i+ξ12i-ω10i2+ω13i2ξ11i-ξ12i,ω12i2=2ξ11iξ12i-ω13i2ξ11i+ω10i2ξ12iξ11i-ξ12i,where ω11i1≠0 and  ω12i1≠0. Then, the compact form of unidirectional auxiliary surface(14)h1=Ω11·e1+Ω12·∫e1+M1,where h1=[h11,h12,h13]T,  ∫e1=[∫e11,∫e12,∫e13]T,  Ω11=diag⁡{ω111,ω121,ω131},  Ω12=diag⁡{ω112,ω122,ω132},   and M1=[M11,M12,M13]T is a constant vector.
As shown in Figure1, the UAS can form a convex set Qi which satisfies Qi⊆Ψi, and the expression of set Qi is written as(15)Qi=e1i,∫e1i∣h1i≥0.The compact form of convex set Qi can be written as Q=[Q1,Q2,Q3]T, where Q⊆Ψ implies Qi⊆Ψi,   i=1,2,3.Figure 1
The convex setQi in attitude constraint Ψi.Step 3.
Similarly, we can define the following switching surfaces for the fast-loop system (8):(16)S21=e2+ξ21·∫e2=0,S22=e2+ξ22·∫e2=0,where ξ21=diag⁡{ξ211,ξ212,ξ213}, ξ22=diag⁡{ξ221,ξ222,ξ223},  S21=[S211,S212,S213]T, S22=[S221,S222,S223]T, and  ξ21i>ξ22i>0,   i=1,2,3.Step 4.
Invoking the coefficientsξ21i,ξ22i in (16), the unidirectional auxiliary surface h2i can be designed as(17)h2i=ω2i1·e2i+ω2i2·∫e2i+M2i,where (18)ω2i1=1S21i<0;S22i<0ω21i1S21i<0;S22i≥0ω22i1S21i≥0;S22i<0-1S21i≥0;S22i≥0;ω2i2=ω20i2S21i<0;S22i<0ω21i2S21i<0;S22i≥0ω22i2S21i≥0;S22i<0-ω23i2S21i≥0;S22i≥0,where ξ22i<ω20i2<ξ21i,  ξ22i<ω23i2<ξ21i, and M2i>0 are the designed parameters, and the rest of coefficients can be designed as follows:(19)ω21i1=-ξ21i+ξ22i+ω20i2+ω23i2ξ21i-ξ22i,ω21i2=-2ξ21iξ22i+ω20i2ξ21i+ω23i2ξ22iξ21i-ξ22i,ω22i1=ξ21i+ξ22i-ω20i2+ω23i2ξ21i-ξ22i,ω21i2=2ξ21iξ22i-ω20i2ξ21i+ω23i2ξ22iξ21i-ξ22i,where ω21i1≠0 and   ω22i1≠0. Then, the compact form of unidirectional auxiliary surface(20)h2=Ω21·e2+Ω22·∫e2+M2,where h2=[h21,h22,h23]T,  ∫e2=[∫e21,∫e22,∫e23]T,  Ω21=diag⁡{ω211,ω221,ω231},  Ω22=diag⁡{ω212,ω222,ω232}, and M2=[M21,M22,M23]T is a constant vector. Then, there exist the following results for the unidirectional auxiliary surfaces h1=[h11,h12,h13]T and  h2=[h21,h22,h23]T.Lemma 2.
For the pointsP1(t)=(e1i,∫e1i) and  P2(t)=(e2i,∫e2i),  i=1,2,3, it follows that δ1i(t)=M1i-h1i(P1)≥0 and δ2i(t)=M2i-h2i(P2)≥0. And if δ1i(t)=0, there exists P1=(0,0); if δ2i(t)=0, there exists P2=(0,0).Proof.
See AppendixA.Lemma 3.
For alle1i,e2i∈R,  i=1,2,3, the functions δ1i(t) and δ2i(t) in Lemma 2 are continuous functions.Proof.
See AppendixB.


## 3.2. Robust Adaptive Backstepping Attitude Control Based on Disturbance Observer and Attitude Constraints

In the attitude control design, we combine the backstepping method, adaptive invariant-set-based sliding mode control, and the disturbance observer technique to design a robust adaptive backstepping attitude controller for the AHV system (1) with attitude constraints and propulsive disturbance, and the detailed design process is appended as follows.


### 3.2.1. Design Process for the Controller

Step 1.
Invoking (14) and (20), we can define δ1 and δ2 as(21)δ1=M1-h1,δ2=M2-h2,where δ1=[δ11,δ12,δ13]T and δ2=[δ21,δ22,δ23]T. Then, the innovative sliding mode surface δ1ε can be defined as(22)δ1ε=δ1-ε1·Satδ1ε1,where δ1ε=[δ1ε1,δ1ε2,δ1ε3]T, ε1=diag⁡{M11,M12,M13}, Sat(δ1/ε1)=[Sat(δ11/M11),Sat(δ12/M12),Sat(δ13/M13)]T, and Sat(·) is the saturation function. From (22), it is clear that(23)δ1εi=δ1i-M1i·Satδ1iM1i=0δ1i≤M1iδ1i-M1iδ1i>M1i.Remark 4.
It is clear thatδ1εi=0 when δ1i=M1i-h1i≤Mi. Considering Qi=e1i,∫e1i∣h1i≥0⊆Ψi in (15), we can obtain that y∈Ψ while δ1ε=0. Therefore, the attitude constraints will be satisfied when system states are sliding on the sliding surface δ1ε=0.Step 2.
In this paper, the nonlinear disturbance observer which is employed to estimated(y,yd,t) is defined as(24)d^=Le2-η,η^˙=Ffx1,x2+Gfx1,x2·u+d^-ν˙1+L-1·Ω21·E,where L=LT>0 is a design parameter which should be chosen to render L-0.5I3×3>0,  E=[1,1,1]T.Assumption 5.
For the estimation errord~=d^-d=[d~1,d~2,d~3]T, there exists positive constant θi such that θi>-τ1i·d~i,  i=1,2,3, where the coefficient τ1i is a bounded element in (4).Step 3.
To suppress the estimation errord~ of disturbance, an adaptive item θ^ is defined as(25)θ^˙=λ·δ1ε,where θ^=[θ^1,θ^2,θ^3]T and λ>0 is a design parameter.Step 4.
Definingθ~=θ^-θ and Δ1ε=diag⁡{δ1ε1,δ1ε2,δ1ε3} and considering (7), the virtual control law ν1 is designed as(26)ν1=Gs-1-Fs+y˙d+Ω11-1-Ω12·e1+N1+Δ1ε·θ^,where N1=[N11,N12,N13]T,  N1i≥0,  i=1,2,3 are designed approaching laws. Using the output of the designed nonlinear disturbance observer (24) and considering Assumptions 3 and 4, the robust attitude control law is designed as(27)u=Gf-1-Ff-d^+ν˙1+Ω21-1-Ω22·e2+N2-Ω21-1Ω11Gse2,where Gf-1 is the generalized matrix inverse of Gf, N2=[N21,N22,N23]T, and  N2i>0.5μi2,  i=1,2,3.Remark 6.
DefiningE=[1,1,1]T and considering Δ1ε=diag⁡{δ1ε1,δ1ε2,δ1ε3} and δ1ε=[δ1ε1,δ1ε2,δ1ε3]T, we can obtain (28)ET·Δ1ε=δ1εT;ET·δ1ε=δ1ε1+δ1ε2+δ1ε3.


### 3.2.2. Proof for the Stability of Sliding Mode Surface

The discussion for the stability of sliding mode surfaceδ1ε=0 is given in this part. Considering δ1ε=0, the control law in (26) can be rewritten as follows:(29)ν1=Gs-1-Fs+y˙d+Ω11-1-Ω12·e1+N1.Substituting (29) into (7) yields(30)e˙1=Ω11-1-Ω12·e1+N1+Gs·e2.Choose the Lyapunov function candidate as(31)V1=ETδ1.From Lemmas 2 and 3, it is clear that the Lyapunov function V1 is a positive and continuous function. And there exists e1=0 while V1=0. Invoking (14) and (21), the time derivative of V1 can be written as(32)V˙1=ETδ˙1=ET-Ω11e˙1-Ω12e1.Substituting (30) into (32) yields(33)V˙1=-ETN1-ETΩ11Gse2.It is apparent that the first term on the right-hand side of (33) is stable. Furthermore, the second term will be canceled as follows.Invoking (4), it is noted that the time derivative of propulsive disturbance d˙1=0 while δ1ε=0. Then, we can obtain d˙=d˙2=τ2 from (2), (3), and (4) and δ1ε=0. Considering the definition of d~ and substituting (27) into (8) yield(34)e˙2=-d~+Ω21-1-Ω22e2+N2-Ω21-1Ω11Gse2.For considering the stability of all signals for the closed-loop control system, the Lyapunov function candidate is chosen as(35)V=V1+0.5d~Td~+ETδ2.The time derivative of V is(36)V˙=V˙1+d~Td~˙+ETδ˙2.Invoking (8), (21), (24), (27), and Assumption 3, the time derivative is(37)V˙=-ETN1-ETN2-d~TLd~-d~d˙≤-ETN1-ETN2-d~TL-0.5I3×3d~+0.5d˙Td˙<-ETN1-ETN2+0.5μTμ-d~TL-0.5I3×3d~,where μ=[μ1,μ2,μ3]T.Theorem 7.
Considering the error systems (7) and (8) with the unknown propulsive disturbance, the disturbance observer is designed as (24) and the robust backstepping attitude control scheme is proposed as (26) and (27). Then, the attitude control error of AHV is convergent when the system states are sliding on the innovative sliding surface δ1ε=0 in (22).Proof.
Invoking (37) and considering L-0.5I3×3>0,  N1i≥0, and N2i>0.5μi2,  i=1,2,3, we have (38)V˙<-ETN1-ETN2+0.5μTμ-d~TL-0.5I3×3d~=-d~TL-0.5I3×3d~-∑N1i+N2i-0.5μi2<0. Thus, the tracking errors e1,e2 and the disturbance approximation error d~ are convergent while the system states are sliding on the innovative sliding mode surface (22).


### 3.2.3. Proof for the Reaching Phase for Sliding Mode Surface

The discussion for the reaching phase for sliding surface is given in this part. Choose the Lyapunov function candidate as(39)V1=ETδ1ε+12λθ~Tθ~.From Lemmas 2 and 3, it is clear that the Lyapunov function V1 is a positive and continuous function, and there exist δ1ε=0 while V1=0. Assume that δ1ε≠0 and considering (23), there exists the following equation:(40)δ˙1ε=δ˙1=-Ω11e˙1-Ω12e1.Substituting (7) and (26) into (40) yields(41)δ˙1ε=-Ω11Gse2-N1-Δ1ε·θ^.Invoking (26) and (39), the time derivative of V1 can be written as(42)V˙1=ETδ˙1ε+θTθ^˙λSubstituting (25), (28), and (40) into (42) yields(43)V˙1=ET-Ω11Gse2-N1-Δ1ε·θ^+δ1εTθ~=ET-Ω11Gse2-N1-δ1εT·θ^+δ1εTθ~.Invoking (2), (4), and (5) yields(44)d˙=d˙1+d˙2=τ1·δ1ε+τ2.Considering the stability of all signals for the closed-loop control system, the Lyapunov function candidate is chosen as(45)V=V1+0.5d~Td~+ETδ2.The time derivative of V is(46)V˙=V˙1+d~Td~˙+ETδ˙2=V˙1+d~Td^˙-d˙+ETδ˙2.Invoking (21), (24), (27), (43), and (44), the time derivative of V is(47)V˙=-ETN1+N2-δ1εTθ^+δ1εTθ~+d~T-Ld~-d~Tτ1·δ1ε+τ2≤-ETN1+N2-δ1εTθ^+δ1εTθ~-d~TL-0.5I3×3d~-d~Tτ1δ1ε+0.5τ2Tτ2.Considering Assumption 5, δ1ε>0, (22), and (23), we have(48)V˙≤-ETN1+N2+δ1εT-θ^+θ~-τ1d~-d~TL-0.5I3×3d~+0.5τ2Tτ2≤-ETN1+N2+δ1εT-θ^+θ~+θ-d~TL-0.5I3×3d~+0.5τ2Tτ2.Substituting θ~=θ^-θ into (48) yields(49)V˙≤-ETN1+N2-d~TL-0.5I3×3d~+0.5τ2Tτ2.Theorem 8.
Considering the attitude motion dynamics (1) of the AHV with attitude constraints and unknown propulsive disturbance, the disturbance observer is designed as (24) and the robust backstepping attitude control scheme is proposed as (26) and (27). Then, attitude controller error of AHV is convergent under the attitude constraints (3). Meanwhile, the system states will arrive at the sliding surface δ1ε=0, and this surface also can be proved as a positively invariant set.Proof.
According to (49), we have(50)V˙≤-d~TL-0.5I3×3d~-∑N1i+N2i-0.5μi2.Substituting N1i≥0,  N2i>0.5μi2, and L-0.5I3×3>0 into (50) yields(51)V˙<0.Thus, the system states will arrive at the sliding surface δ1ε=0. From Remark 4, it is noted that attitude constraints (3) will be satisfied when system states are sliding on sliding surface δ1ε=0.
Considering the stability of sliding surface as shown in Theorem8, the attitude control error of AHV is convergent under the attitude constraints (3). From (39), (45), and (51), there exists δ1ε(t)=0 for all t>t1 if V(t1)=0. And the sliding surface δ1ε(t)=0 can be rewritten as a closed set Q as shown in (15) and Remark 4. Then, the sliding surface δ1ε=0 can be seen as a positively invariant set according to Definition 1.


## 3.2.1. Design Process for the Controller

Step 1.
Invoking (14) and (20), we can define δ1 and δ2 as(21)δ1=M1-h1,δ2=M2-h2,where δ1=[δ11,δ12,δ13]T and δ2=[δ21,δ22,δ23]T. Then, the innovative sliding mode surface δ1ε can be defined as(22)δ1ε=δ1-ε1·Satδ1ε1,where δ1ε=[δ1ε1,δ1ε2,δ1ε3]T, ε1=diag⁡{M11,M12,M13}, Sat(δ1/ε1)=[Sat(δ11/M11),Sat(δ12/M12),Sat(δ13/M13)]T, and Sat(·) is the saturation function. From (22), it is clear that(23)δ1εi=δ1i-M1i·Satδ1iM1i=0δ1i≤M1iδ1i-M1iδ1i>M1i.Remark 4.
It is clear thatδ1εi=0 when δ1i=M1i-h1i≤Mi. Considering Qi=e1i,∫e1i∣h1i≥0⊆Ψi in (15), we can obtain that y∈Ψ while δ1ε=0. Therefore, the attitude constraints will be satisfied when system states are sliding on the sliding surface δ1ε=0.Step 2.
In this paper, the nonlinear disturbance observer which is employed to estimated(y,yd,t) is defined as(24)d^=Le2-η,η^˙=Ffx1,x2+Gfx1,x2·u+d^-ν˙1+L-1·Ω21·E,where L=LT>0 is a design parameter which should be chosen to render L-0.5I3×3>0,  E=[1,1,1]T.Assumption 5.
For the estimation errord~=d^-d=[d~1,d~2,d~3]T, there exists positive constant θi such that θi>-τ1i·d~i,  i=1,2,3, where the coefficient τ1i is a bounded element in (4).Step 3.
To suppress the estimation errord~ of disturbance, an adaptive item θ^ is defined as(25)θ^˙=λ·δ1ε,where θ^=[θ^1,θ^2,θ^3]T and λ>0 is a design parameter.Step 4.
Definingθ~=θ^-θ and Δ1ε=diag⁡{δ1ε1,δ1ε2,δ1ε3} and considering (7), the virtual control law ν1 is designed as(26)ν1=Gs-1-Fs+y˙d+Ω11-1-Ω12·e1+N1+Δ1ε·θ^,where N1=[N11,N12,N13]T,  N1i≥0,  i=1,2,3 are designed approaching laws. Using the output of the designed nonlinear disturbance observer (24) and considering Assumptions 3 and 4, the robust attitude control law is designed as(27)u=Gf-1-Ff-d^+ν˙1+Ω21-1-Ω22·e2+N2-Ω21-1Ω11Gse2,where Gf-1 is the generalized matrix inverse of Gf, N2=[N21,N22,N23]T, and  N2i>0.5μi2,  i=1,2,3.Remark 6.
DefiningE=[1,1,1]T and considering Δ1ε=diag⁡{δ1ε1,δ1ε2,δ1ε3} and δ1ε=[δ1ε1,δ1ε2,δ1ε3]T, we can obtain (28)ET·Δ1ε=δ1εT;ET·δ1ε=δ1ε1+δ1ε2+δ1ε3.


## 3.2.2. Proof for the Stability of Sliding Mode Surface

The discussion for the stability of sliding mode surfaceδ1ε=0 is given in this part. Considering δ1ε=0, the control law in (26) can be rewritten as follows:(29)ν1=Gs-1-Fs+y˙d+Ω11-1-Ω12·e1+N1.Substituting (29) into (7) yields(30)e˙1=Ω11-1-Ω12·e1+N1+Gs·e2.Choose the Lyapunov function candidate as(31)V1=ETδ1.From Lemmas 2 and 3, it is clear that the Lyapunov function V1 is a positive and continuous function. And there exists e1=0 while V1=0. Invoking (14) and (21), the time derivative of V1 can be written as(32)V˙1=ETδ˙1=ET-Ω11e˙1-Ω12e1.Substituting (30) into (32) yields(33)V˙1=-ETN1-ETΩ11Gse2.It is apparent that the first term on the right-hand side of (33) is stable. Furthermore, the second term will be canceled as follows.Invoking (4), it is noted that the time derivative of propulsive disturbance d˙1=0 while δ1ε=0. Then, we can obtain d˙=d˙2=τ2 from (2), (3), and (4) and δ1ε=0. Considering the definition of d~ and substituting (27) into (8) yield(34)e˙2=-d~+Ω21-1-Ω22e2+N2-Ω21-1Ω11Gse2.For considering the stability of all signals for the closed-loop control system, the Lyapunov function candidate is chosen as(35)V=V1+0.5d~Td~+ETδ2.The time derivative of V is(36)V˙=V˙1+d~Td~˙+ETδ˙2.Invoking (8), (21), (24), (27), and Assumption 3, the time derivative is(37)V˙=-ETN1-ETN2-d~TLd~-d~d˙≤-ETN1-ETN2-d~TL-0.5I3×3d~+0.5d˙Td˙<-ETN1-ETN2+0.5μTμ-d~TL-0.5I3×3d~,where μ=[μ1,μ2,μ3]T.Theorem 7.
Considering the error systems (7) and (8) with the unknown propulsive disturbance, the disturbance observer is designed as (24) and the robust backstepping attitude control scheme is proposed as (26) and (27). Then, the attitude control error of AHV is convergent when the system states are sliding on the innovative sliding surface δ1ε=0 in (22).Proof.
Invoking (37) and considering L-0.5I3×3>0,  N1i≥0, and N2i>0.5μi2,  i=1,2,3, we have (38)V˙<-ETN1-ETN2+0.5μTμ-d~TL-0.5I3×3d~=-d~TL-0.5I3×3d~-∑N1i+N2i-0.5μi2<0. Thus, the tracking errors e1,e2 and the disturbance approximation error d~ are convergent while the system states are sliding on the innovative sliding mode surface (22).


## 3.2.3. Proof for the Reaching Phase for Sliding Mode Surface

The discussion for the reaching phase for sliding surface is given in this part. Choose the Lyapunov function candidate as(39)V1=ETδ1ε+12λθ~Tθ~.From Lemmas 2 and 3, it is clear that the Lyapunov function V1 is a positive and continuous function, and there exist δ1ε=0 while V1=0. Assume that δ1ε≠0 and considering (23), there exists the following equation:(40)δ˙1ε=δ˙1=-Ω11e˙1-Ω12e1.Substituting (7) and (26) into (40) yields(41)δ˙1ε=-Ω11Gse2-N1-Δ1ε·θ^.Invoking (26) and (39), the time derivative of V1 can be written as(42)V˙1=ETδ˙1ε+θTθ^˙λSubstituting (25), (28), and (40) into (42) yields(43)V˙1=ET-Ω11Gse2-N1-Δ1ε·θ^+δ1εTθ~=ET-Ω11Gse2-N1-δ1εT·θ^+δ1εTθ~.Invoking (2), (4), and (5) yields(44)d˙=d˙1+d˙2=τ1·δ1ε+τ2.Considering the stability of all signals for the closed-loop control system, the Lyapunov function candidate is chosen as(45)V=V1+0.5d~Td~+ETδ2.The time derivative of V is(46)V˙=V˙1+d~Td~˙+ETδ˙2=V˙1+d~Td^˙-d˙+ETδ˙2.Invoking (21), (24), (27), (43), and (44), the time derivative of V is(47)V˙=-ETN1+N2-δ1εTθ^+δ1εTθ~+d~T-Ld~-d~Tτ1·δ1ε+τ2≤-ETN1+N2-δ1εTθ^+δ1εTθ~-d~TL-0.5I3×3d~-d~Tτ1δ1ε+0.5τ2Tτ2.Considering Assumption 5, δ1ε>0, (22), and (23), we have(48)V˙≤-ETN1+N2+δ1εT-θ^+θ~-τ1d~-d~TL-0.5I3×3d~+0.5τ2Tτ2≤-ETN1+N2+δ1εT-θ^+θ~+θ-d~TL-0.5I3×3d~+0.5τ2Tτ2.Substituting θ~=θ^-θ into (48) yields(49)V˙≤-ETN1+N2-d~TL-0.5I3×3d~+0.5τ2Tτ2.Theorem 8.
Considering the attitude motion dynamics (1) of the AHV with attitude constraints and unknown propulsive disturbance, the disturbance observer is designed as (24) and the robust backstepping attitude control scheme is proposed as (26) and (27). Then, attitude controller error of AHV is convergent under the attitude constraints (3). Meanwhile, the system states will arrive at the sliding surface δ1ε=0, and this surface also can be proved as a positively invariant set.Proof.
According to (49), we have(50)V˙≤-d~TL-0.5I3×3d~-∑N1i+N2i-0.5μi2.Substituting N1i≥0,  N2i>0.5μi2, and L-0.5I3×3>0 into (50) yields(51)V˙<0.Thus, the system states will arrive at the sliding surface δ1ε=0. From Remark 4, it is noted that attitude constraints (3) will be satisfied when system states are sliding on sliding surface δ1ε=0.
Considering the stability of sliding surface as shown in Theorem8, the attitude control error of AHV is convergent under the attitude constraints (3). From (39), (45), and (51), there exists δ1ε(t)=0 for all t>t1 if V(t1)=0. And the sliding surface δ1ε(t)=0 can be rewritten as a closed set Q as shown in (15) and Remark 4. Then, the sliding surface δ1ε=0 can be seen as a positively invariant set according to Definition 1.


## 4. Simulation Results

In this section, simulation results are given to illustrate the effectiveness of the proposed adaptive UAS-SMC schemes for AHV with attitude constraints. Suppose that the AHV vehicle lies in the cruise flight phase with the velocity 3000 m/s and flight altitude 30 km. The initial attitude and attitude angular velocity conditions are chosen asα0=0deg, β0=0 deg,  μ0=0 deg, and p0=q0=r0=0 deg/s. The attitude constraints for the state errors e1=y-yd=[e11,e12,e13]T are given as(52)-1deg≤e11≤1deg,-0.2deg≤e12≤0.2deg,-1deg≤e13≤1deg. In the simulation, we assume that the unknown time-varying disturbance moments imposed on the AHV are(53)dy,yd,t=d1y,yd,t+d2t,where d1=106diag⁡{gpl,gqm,grn}d1m;  d2=106diag⁡{gpl,gqm,grn}d2m;  d˙1m=sin⁡(20α)·δ1ε;  d2m=[0.2sin⁡πt,3sin⁡πt,sin⁡(πt)]T, and the coefficients gpl, gqm, and grn can be found in [24].According to the design steps in Section3, the adaptive UAS-SMC control design parameters are chosen as ξ11=diag⁡{5,5,5},  ξ12=diag⁡{1,1,1},  ξ21=diag⁡{5,5,5}, ξ22=diag⁡{2,2,2},  L=diag⁡{2,2,2},  λ=1, and  ε1=diag⁡{0.02,0.02,0.02};  ω10i2=ω13i2=5/3 and  ω20i2=ω23i2=3,   i=1,2,3. The detailed approaching laws N1>0,  N2>0 can be found in [25], and the innovative sliding mode surface δ1ε can be defined as (54)δ1ε=δ1-ε1·Satδ1ε1,θ^˙=λ·δ1ε.Then, the simulation results are given as in Figures 2–5.Figure 2
The attitude responses of AHV system under SMC and UAS-SMC methods.Figure 3
The state error responses under SMC + NDO and adaptive UAS-SMC + NDO.Figure 4
The disturbance estimations under NDO [3] and UAS-NDO with the same coefficient L.Figure 5
Adaptive UAS-SMC + NDO attitude control input.The attitude and state error responses are shown in Figures2–5 under SMC and adaptive UAS-SMC methods, respectively. From Figure 2, attack angle α in system (1) is unstable when propulsive disturbance d1 is not considered in the design process of SMC controller. To handle this problem, the conservative parameters and nonlinear disturbance observer are given in the SMC control design as shown in Figure 3. However, the undesirable overshoots are often found with conservative parameters since attitude constraints have not been considered in the design process of SMC scheme. And these overshoots are harmful for the AHV system because they might be out of the attitude constraints as shown in Figure 3. From the state error responses in Figure 4, the attitude constraints can be satisfied with adaptive UAS-SMC method. And harmful overshoots will be removed by the designed positively invariant sets. Therefore, we know that the proposed adaptive UAS-SMC control scheme can efficiently track the desired trajectories with attitude constraints and propulsive disturbance. Figure 4 shows the disturbance estimation under NDO [3] and UAS-NDO with the same coefficient L. Unlike the previous NDO, the switching matrix Ω21 of UAS is introduced in the UAS-NDO. This matrix will improve the performance of disturbance observer (24). Then, we can obtain a new fast-nonlinear disturbance observer for the time-varying propulsive disturbance of AHV systems.


## 5. Conclusion

In this paper, the adaptive UAS-SMC controller with nonlinear disturbance observer has been proposed for the AHV with the unknown propulsive disturbance and the attitude constraints. To handle the propulsive disturbance, a developed fast-nonlinear disturbance observer is proposed to estimate the propulsive disturbance using adaptive method. And an innovative sliding surface is firstly proposed for the attitude constraints with invariant set theory. Rigorous analysis has been given for the convergence of all closed-loop signals under the proposed control schemes. Simulation results show the effectiveness of the robust adaptive UAS-SMC scheme for the AHV. In the following study, the robust attitude control scheme can be further developed for the AHV with time-varying attitude constraints.



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*Source: 293480-2015-12-27.xml*