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

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1 Mathematical Problems in Engineering Volume 215 Article ID pages Research Article Robust Adaptive Attitude Control for Airbreathing Hypersonic Vehicle with Attitude Constraints and Propulsive Disturbance Jian Fu 1 Liangming Wang 1 Mou Chen 2 and Sijiang Chang 1 1 College of Energy and Power Engineering Nanjing University of Science and Technology Nanjing 2194 China 2 College of Automation Engineering Nanjing University of Aeronautics and Astronautics Nanjing 2116 China Correspondence should be addressed to Jian Fu; fujian@njust.edu.cn Received 16 July 215; Accepted 26 November 215 Academic Editor: Marek Lefik Copyright 215 Jian Fu et al. This is an open access article distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. A robust adaptive backstepping attitude control scheme combined with invariant-set-based sliding mode control and fastnonlinear 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. 1. Introduction In recent years a new kind of aerospace vehicle which is called airbreathing hypersonic vehicle (AHV) has attracted theconsiderableattentioninbothmilitaryandcivilcommunities. 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 havefocusedonthedisturbanceobserverofahvflight 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 [1 11]. Thus this kind of disturbance will lead to a challenging problem for

2 2 Mathematical Problems in Engineering 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. Section 2 details the problem formulation while main results are given in Section 3. Simulation results are presentedinsection4toshowtheeffectivenessofproposed robust adaptive backstepping attitude control for AHV. And some conclusion results are shown in Section 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]: x 1 = F s (x 1 )+G s (x 1 ) x 2 x 2 = F f (x 1 x 2 )+G f (x 1 x 2 ) u + d (t) y = x 1 where x 1 =[αβμ] T R 3 is the attitude angle vector of slowloop states x 2 =[pqr] T R 3 is the body-axis angular rate vector of fast-loop states u =[δ e δ a δ r δ x δ y δ z ] T R 6 is (1) the deflection vector of control surfaces which are the system control inputs y R 3 isthesystemoutputandtheunknown disturbance d(y y d t) R 3 is defined as d (y y d t)=d 1 (y y d t)+d 2 (t) (2) where d 1 (y y d t) R 3 is the unknown propulsive disturbance which is coupling with the attitude angles and d 2 (t) R 3 is the rest of unknown disturbance. The detailed expression of F s R 3 G s R 3 3 F f R 3 andg f R 3 6 can be found in [24] Assumptions and Control Objective. In order to reduce the coupling effects between the scramjet propulsion system andattitudeanglestheattitudeconstraintsaredefinedas Ψ =y R 3 y (i) d =123} c i y i y (i) d +c i c i R + i where y = [y 1 y 2 y 3 ] T is the attitude angle vector and y d = [y (1) d y(2) d y(3) d ]T is the desired output signal. When the attitude angle vector y satisfies constraints (3) the propulsive disturbance d 1 (yt) 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 d 1 (yt) and d 2 (t) will satisfy the following assumptions. Assumption 1. For all t > there exists Δ i such that y (i) d (t) Δ i i = Assumption 2. The propulsive disturbance d 1 (y y d t)can be described as the following differential equation: d 1 (y y d t)=τ 1 (yt) δ 1ε (y y d ) (4) where τ 1 (yt) = diagτ 11 τ 12 τ 13 } and we can obtain τ 1 (yt)=[] T if y ψ.fortheelementτ 1i there exists an unknown positive constant κ i such that τ 1i κ i i = The vector δ 1ε R 3 is a special designed sliding surface as shown in (22) and there exists y Ψ while δ 1ε =. Assumption 3. The disturbance d 2 (t) can be described as the following differential equation: d 2 (t) = τ 2 (t) (5) where τ 2 (t) = [τ 21 τ 22 τ 23 ] T and there exists a known positive constant μ i such that τ 2i (t) μ i i = Assumption 4. The generalized matrix inverses of G s and G f are always existing for the nonlinear attitude motion model of the AHV and the elements in matrixes F s G s F f andg f are always continuous for the nonlinear attitude motion model of the AHV. Definition 1 (see [25]). The set Q is said to be positively invariant (PI) for nonlinear system (1) if for all x 1 (t 1 ) Q there exists the solution x 1 (t 2 ) Q t 2 >t 1. (3)

3 Mathematical Problems in Engineering 3 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 e 1 = y y d e 2 = x 2 ^1 where ^1 R 3 is a designed virtual control law. Then considering (1) the derivatives e 1 e 2 can be written as (6) e 1 = F s (x 1 ) + G s (x 1 ) (e 2 + ^1) y d (7) e 2 = F f (x 1 x 2 )+G f (x 1 x 2 ) u + d (y y d t) ^1 (8) where e 1 =[e 11 e 12 e 13 ] T R 3 are the state errors of slowloop states and e 2 =[e 21 e 22 e 23 ] T R 3 arethestateerrorsof fast-loop states. Invoking (6) the attitude constraints (3) can be rewritten as Ψ =(e 1 e 1 ) c i e 1i c i c i R + } (9) where Ψ =[Ψ 1 Ψ 2 Ψ 3 ] T and Ψ i =(e 1i e 1i ) c i e 1i c i }. Considering the attitude constraints of AHV the unidirectional auxiliary surfaces (UAS) for slow-loop and fastloop 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 S 11 = e 1 + ξ 11 e 1 = S 12 = e 1 + ξ 12 e 1 = (1) where ξ 11 = diagξ 111 ξ 112 ξ 113 } ξ 12 = diagξ 121 ξ 122 ξ 123 } S 11 =[S 111 S 112 S 113 ] T S 12 =[S 121 S 122 S 123 ] T ξ 11i > ξ 12i > i = and e 1 dt is denoted by e 1 for brevity. The conditions ξ 11i >and ξ 12i >are given to guarantee the stability of switching surfaces S 11i =and S 12i =.Andthe condition ξ 11i >ξ 12i is used to avoid the overlap of switching surfaces. Step 2. Invoking the coefficients ξ 11i ξ 12i in (1) the unidirectional auxiliary surface h 1i canbedesignedas h 1i =ω 1i1 e 1i +ω 1i2 e 1i +M 1i (11) where 1 S 11i <; S 12i < ω 11i1 S 11i <; S 12i ω 1i1 = ω 12i1 S 11i ; S 12i < 1 S 11i ; S 12i ; ω 1i2 S 11i <; S 12i < ω 11i2 S 11i <; S 12i ω 1i2 = ω 12i2 S 11i ; S 12i < ω 13i2 S 11i ; S 12i (12) where ω 1i2 ω 13i2 are the designed parameters which satisfy ξ 12i <ω 1i2 <ξ 11i ξ 12i <ω 13i2 <ξ 11i andm 1i >andthe rest of coefficients are given as follows: ω 11i1 = [ (ξ 11i +ξ 12i )+(ω 1i2 +ω 13i2 )] ω 11i2 = [ 2ξ 11iξ 12i +(ω 1i2 ξ 11i +ω 13i2 ξ 12i )] ω 12i1 = [(ξ 11i +ξ 12i ) (ω 1i2 +ω 13i2 )] ω 12i2 = [2ξ 11iξ 12i (ω 13i2 ξ 11i +ω 1i2 ξ 12i )] (13) where ω 11i1 =and ω 12i1 =.Thenthecompactformof unidirectional auxiliary surface h 1 = Ω 11 e 1 + Ω 12 e 1 + M 1 (14) where h 1 = [h 11 h 12 h 13 ] T e 1 = [ e 11 e 12 e 13 ] T Ω 11 = diagω 111 ω 121 ω 131 } Ω 12 = diagω 112 ω 122 ω 132 } and M 1 =[M 11 M 12 M 13 ] T is a constant vector. As shown in Figure 1 the UAS can form a convex set Q i which satisfies Q i Ψ i and the expression of set Q i is written as Q i =(e 1i e 1i ) h 1i }. (15) ThecompactformofconvexsetQ i can be written as Q = [Q 1 Q 2 Q 3 ] T whereq Ψ implies Q i Ψ i i = Step 3. Similarly we can define the following switching surfaces for the fast-loop system (8): S 21 = e 2 + ξ 21 e 2 = S 22 = e 2 + ξ 22 e 2 = (16) where ξ 21 = diagξ 211 ξ 212 ξ 213 } ξ 22 = diagξ 221 ξ 222 ξ 223 } S 21 = [S 211 S 212 S 213 ] T S 22 = [S 221 S 222 S 223 ] T and ξ 21i > ξ 22i > i =

4 4 Mathematical Problems in Engineering S 11i S 12i < h 12i e 1i = c i h 1i S 11i <S 12i < e 1i Ψ i = (e 1i e 1i ) c i e 1i c i } S 11i S 12i Q i = (e 1i e 1i ) h 1i } h 13i h 11i e 1i =c i S 11i <S 12i S 12i Figure 1: The convex set Q i in attitude constraint Ψ i. Step 4. Invoking the coefficients ξ 21i ξ 22i in (16) the unidirectional auxiliary surface h 2i canbedesignedas where e 1i S 11i h 2i =ω 2i1 e 2i +ω 2i2 e 2i +M 2i (17) 1 S 21i <; S 22i < ω 21i1 S 21i <; S 22i ω 2i1 = ω 22i1 S 21i ; S 22i < 1 S 21i ; S 22i ; ω 2i2 S 21i <; S 22i < ω 21i2 S 21i <; S 22i ω 2i2 = ω 22i2 S 21i ; S 22i < ω 23i2 S 21i ; S 22i (18) where ξ 22i <ω 2i2 <ξ 21i ξ 22i <ω 23i2 <ξ 21i andm 2i >are the designed parameters and the rest of coefficients can be designed as follows: ω 21i1 = [ (ξ 21i +ξ 22i )+(ω 2i2 +ω 23i2 )] (ξ 21i ξ 22i ) where ω 21i1 =and ω 22i1 =. Then the compact form of unidirectional auxiliary surface h 2 = Ω 21 e 2 + Ω 22 e 2 + M 2 (2) where h 2 = [h 21 h 22 h 23 ] T e 2 = [ e 21 e 22 e 23 ] T Ω 21 = diagω 211 ω 221 ω 231 } Ω 22 = diagω 212 ω 222 ω 232 } and M 2 =[M 21 M 22 M 23 ] T is a constant vector. Then there exist the following results for the unidirectional auxiliary surfaces h 1 =[h 11 h 12 h 13 ] T and h 2 =[h 21 h 22 h 23 ] T. Lemma 2. For the points P 1 (t) = (e 1i e 1i ) and P 2 (t) = (e 2i e 2i ) i = 1 2 3itfollowsthatδ 1i (t) = M 1i h 1i (P 1 ) and δ 2i (t) = M 2i h 2i (P 2 ).Andifδ 1i (t) = there exists P 1 =();ifδ 2i (t) = there exists P 2 = ( ). Proof. See Appendix A. Lemma 3. For all e 1i e 2i R i = thefunctionsδ 1i (t) and δ 2i (t) in Lemma 2 are continuous functions. Proof. See Appendix B 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 Design Process for the Controller Step 1. Invoking (14) and (2) we can define δ 1 and δ 2 as δ 1 = M 1 h 1 δ 2 = M 2 h 2 (21) where δ 1 =[δ 11 δ 12 δ 13 ] T and δ 2 =[δ 21 δ 22 δ 23 ] T.Thenthe innovative sliding mode surface δ 1ε can be defined as δ 1ε = δ 1 ε 1 Sat ( δ 1 ε 1 ) (22) where δ 1ε = [δ 1ε1 δ 1ε2 δ 1ε3 ] T ε 1 = diagm 11 M 12 M 13 } Sat(δ 1 /ε 1 ) = [Sat(δ 11 /M 11 ) Sat(δ 12 /M 12 ) Sat(δ 13 /M 13 )] T and Sat( ) is the saturation function. From (22) it is clear that ω 21i2 = [ 2ξ 21iξ 22i +(ω 2i2 ξ 21i +ω 23i2 ξ 22i )] (ξ 21i ξ 22i ) ω 22i1 = [(ξ 21i +ξ 22i ) (ω 2i2 +ω 23i2 )] (ξ 21i ξ 22i ) ω 21i2 = [2ξ 21iξ 22i (ω 2i2 ξ 21i +ω 23i2 ξ 22i )] (ξ 21i ξ 22i ) (19) δ 1εi =δ 1i M 1i Sat ( δ 1i M 1i ) = δ 1i M 1i δ 1i M 1i δ 1i >M 1i. (23) Remark 4. It is clear that δ 1εi =when δ 1i =M 1i h 1i M i. Considering Q i = (e 1i e 1i ) h 1i } Ψ i in (15) we

5 Mathematical Problems in Engineering 5 can obtain that y Ψ while δ 1ε =. Therefore the attitude constraints will be satisfied when system states are sliding on the sliding surface δ 1ε =. Step 2. In this paper the nonlinear disturbance observer whichisemployedtoestimated(y y d t)is defined as d = L (e 2 η) η = F f (x 1 x 2 )+G f (x 1 x 2 ) u + d ^1 + L 1 Ω 21 E (24) where L = L T > is a design parameter which should be chosen to render L.5I 3 3 > E = [1 1 1] T. Assumption 5. For the estimation error d = d d = [ d 1 d 2 d 3 ] T there exists positive constant θ i such that θ i > τ 1i di i = wherethecoefficientτ 1i is a bounded element in (4). Step 3. To suppress the estimation error d of disturbance an adaptive item θ is defined as θ =λ δ 1ε (25) where θ =[ θ 1 θ 2 θ 3 ] T and λ>is a design parameter. Step 4. Defining θ = θ θ and Δ 1ε = diagδ 1ε1 δ 1ε2 δ 1ε3 } and considering (7) the virtual control law ^1 is designed as ^1 = G 1 s [ F s + y d + Ω 1 11 ( Ω 12 e 1 + N 1 + Δ 1ε θ)] (26) where N 1 = [N 11 N 12 N 13 ] T N 1i i = 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 u = G 1 f ( F f d + ^1 + Ω 1 21 ( Ω 22 e 2 + N 2 ) Ω 1 21 Ω 11G s e 2 ) (27) where G 1 f is the generalized matrix inverse of G f N 2 = [N 21 N 22 N 23 ] T and N 2i >.5μ 2 i i=123. Remark 6. Defining E = [1 1 1] T and considering Δ 1ε = diagδ 1ε1 δ 1ε2 δ 1ε3 } and δ 1ε = [δ 1ε1 δ 1ε2 δ 1ε3 ] T wecan obtain E T Δ 1ε = δ T 1ε ; E T δ 1ε =δ 1ε1 +δ 1ε2 +δ 1ε3. (28) Proof for the Stability of Sliding Mode Surface. The discussion for the stability of sliding mode surface δ 1ε = is given in this part. Considering δ 1ε =thecontrollawin (26) can be rewritten as follows: ^1 = G 1 s ( F s + y d + Ω 1 11 ( Ω 12 e 1 + N 1 )). (29) Substituting (29) into (7) yields e 1 = Ω 1 11 ( Ω 12 e 1 + N 1 )+G s e 2. (3) Choose the Lyapunov function candidate as V 1 = E T δ 1. (31) From Lemmas 2 and 3 it is clear that the Lyapunov function V 1 is a positive and continuous function. And there exists e 1 = while V 1 =. Invoking (14) and (21) the time derivative of V 1 canbewrittenas V 1 = E T δ 1 = E T ( Ω 11 e 1 Ω 12 e 1 ). (32) Substituting (3) into (32) yields V 1 = E T N 1 E T Ω 11 G s e 2. (33) 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 =while δ 1ε =.Thenwecan obtain d = d 2 = τ 2 from (2) (3) and (4) and δ 1ε =. Considering the definition of d and substituting (27) into (8) yield e 2 = d + Ω 1 21 ( Ω 22e 2 + N 2 ) Ω 1 21 Ω 11G s e 2. (34) For considering the stability of all signals for the closed-loop control system the Lyapunov function candidate is chosen as The time derivative of V is V=V d T d + E T δ 2. (35) V= V 1 T + d d + E T δ 2. (36) Invoking (8) (21) (24) (27) and Assumption 3 the time derivative is V= E T N 1 E T N 2 d T L d d d E T N 1 E T N 2 d T (L.5I 3 3 ) d +.5 d T d < E T N 1 E T N 2 +.5μ T μ d T (L.5I 3 3 ) d where μ =[μ 1 μ 2 μ 3 ] T. (37) 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 δ 1ε =in (22).

6 6 Mathematical Problems in Engineering Proof. Invoking (37) and considering L.5I 3 3 > N 1i andn 2i >.5μ 2 i i=123wehave V< E T N 1 E T N 2 +.5μ T μ d T (L.5I 3 3 ) d = d T (L.5I 3 3 ) d (N 1i +N 2i.5μ 2 i ) <. (38) Thus the tracking errors e 1 e 2 andthedisturbanceapproximation error d are convergent while the system states are sliding on the innovative sliding mode surface (22) Proof for the Reaching Phase for Sliding Mode Surface. The discussion for the reaching phase for sliding surface is giveninthispart.choosethelyapunovfunctioncandidate as V 1 = E T δ 1ε + 1 2λ θ T θ. (39) From Lemmas 2 and 3 it is clear that the Lyapunov function V 1 is a positive and continuous function and there exist δ 1ε = while V 1 =. Assume that δ 1ε =and considering (23) there exists the following equation: δ 1ε = δ 1 = Ω 11 e 1 Ω 12 e 1. (4) Substituting (7) and (26) into (4) yields δ 1ε = Ω 11 G s e 2 N 1 Δ 1ε θ. (41) Invoking (26) and (39) the time derivative of V 1 can be written as V 1 = E T δ 1ε + θt θ (42) λ Substituting (25) (28) and (4) into (42) yields V 1 = E T ( Ω 11 G s e 2 N 1 Δ 1ε θ)+δ T 1ε θ = E T ( Ω 11 G s e 2 N 1 ) δ T 1ε θ (43) + δ T 1ε θ. Invoking (2) (4) and (5) yields d = d 1 + d 2 = τ 1 δ 1ε + τ 2. (44) Considering the stability of all signals for the closed-loop control system the Lyapunov function candidate is chosen as V=V d T d + E T δ 2. (45) The time derivative of V is V = V 1 T + d d + E T δ 2 = V 1 + d T ( d d) + E T δ 2. (46) Invoking (21) (24) (27) (43) and (44) the time derivative of V is V= E T (N 1 + N 2 ) δ T 1ε θ + δ T 1ε θ + d T ( L) d d T (τ 1 δ 1ε + τ 2 ) E T (N 1 + N 2 ) δ T 1ε θ + δ T 1ε θ d T (L.5I 3 3 ) d d T τ 1 δ 1ε +.5τ T 2 τ 2. (47) Considering Assumption 5 δ 1ε >(22)and(23)wehave V E T (N 1 + N 2 )+δ T 1ε [ θ + θ τ 1 d] d T (L.5I 3 3 ) d +.5τ T 2 τ 2 E T (N 1 + N 2 )+δ T 1ε [ θ + θ + θ] d T (L.5I 3 3 ) d +.5τ T 2 τ 2. Substituting θ = θ θ into (48) yields (48) V E T (N 1 + N 2 ) d T (L.5I 3 3 ) d +.5τ T 2 τ 2. (49) 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 therobustbacksteppingattitudecontrolschemeisproposed 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 δ 1ε =andthis surfacealsocanbeprovedasapositivelyinvariantset. Proof. According to (49) we have V d T (L.5I 3 3 ) d (N 1i +N 2i.5μ 2 i ). (5) Substituting N 1i N 2i >.5μ 2 i andl.5i 3 3 >into (5) yields V <. (51) Thus the system states will arrive at the sliding surface δ 1ε =. From Remark 4 it is noted that attitude constraints (3) will be satisfied when system states are sliding on sliding surface δ 1ε =. Considering the stability of sliding surface as shown in Theorem 8 the attitude control error of AHV is convergent under the attitude constraints (3). From (39) (45) and (51) there exists δ 1ε (t) = for all t>t 1 if V(t 1 )=.Andthe sliding surface δ 1ε (t) = canberewrittenasaclosedset Q as shown in (15) and Remark 4. Then the sliding surface δ 1ε =can be seen as a positively invariant set according to Definition Simulation Results In this section simulation results are given to illustrate the effectivenessoftheproposedadaptiveuas-smcschemesfor AHV with attitude constraints. Suppose that the AHV vehicle liesinthecruiseflightphasewiththevelocity3m/sand flight altitude 3 km. The initial attitude and attitude angular velocity conditions are chosen as α =deg β =deg μ = deg and p =q =r =deg/s. The attitude constraints for the state errors e 1 = y y d =[e 11 e 12 e 13 ] T are given as 1 deg e 11 1deg.2 deg e 12.2 deg 1 deg e 13 1deg. (52)

7 Mathematical Problems in Engineering 7 In the simulation we assume that the unknown time-varying disturbancemomentsimposedontheahvare d (y y d t) = d 1 (y y d t) + d 2 (t) (53) where d 1 =1 6 diagg l p gm q gn r }d 1m; d 2 =1 6 diagg l p gm q g n r }d 2m; d 1m = sin(2α) δ 1ε ; d 2m = [.2 sin(πt) 3 sin(πt) sin(πt)] T and the coefficients g l p gm q andgn r can be found in [24]. According to the design steps in Section 3 the adaptive UAS-SMC control design parameters are chosen as ξ 11 = diag5 5 5} ξ 12 = diag1 1 1} ξ 21 = diag555} ξ 22 = diag222} L = diag2 2 2} λ = 1andε 1 = diag.2.2.2}; ω 1i2 =ω 13i2 =5/3and ω 2i2 =ω 23i2 = 3 i = The detailed approaching laws N 1 > N 2 > can be found in [25] and the innovative sliding mode surface δ 1ε canbedefinedas α (deg) Reference signal Adaptive UAS-SMC + NDO SMC Figure 2: The attitude responses of AHV system under SMC and UAS-SMC methods. δ 1ε = δ 1 ε 1 Sat ( δ 1 ) ε 1 θ =λ δ 1ε. (54) 4 3 Then the simulation results are given as in Figures 2 5. The attitude and state error responses are shown in Figures 2 5 under SMC and adaptive UAS-SMC methods respectively. From Figure 2 attack angle α in system (1) is unstable when propulsive disturbance d 1 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 theproposedadaptiveuas-smccontrolschemecanefficiently 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.UnlikethepreviousNDOtheswitchingmatrix Ω 21 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 e 12 (deg) e 11 (deg) e 13 (deg) Attitude constraints e SMC-NDO Adaptive UAS-SMC + NDO SMC-NDO Adaptive UAS-SMC + NDO Attitude constraints e SMC-NDO Adaptive UAS-SMC + NDO Attitude constraints e 11 1 Figure 3: The state error responses under SMC + NDO and adaptive UAS-SMC + NDO.

8 8 Mathematical Problems in Engineering d 13 +d 23 d 12 +d 22 d 11 +d Disturbance UAS-NDO NDO Disturbance UAS-NDO NDO Disturbance UAS-NDO NDO Figure 4: The disturbance estimations under NDO [3] and UAS- NDO with the same coefficient L. the attitude constraints with invariant set theory. Rigorous analysis has been given for the convergence of all closedloop signals under the proposed control schemes. Simulation resultsshowtheeffectivenessoftherobustadaptiveuas- SMCschemefortheAHV.Inthefollowingstudytherobust attitude control scheme can be further developed for the AHV with time-varying attitude constraints. Appendices A. Proof of Lemma 2 Take the point P 1 (t) = (e 1i e 1i ) and δ 1i (t) = M 1i h 1i (P 1 ) as examples; we have the following discussions. From the definition of h 1i in (11) the function δ 1i (t) is switching among the following subspaces: No. 1i Subspace: S 11i < S 12i < No. 1 1i Subspace: S 11i < S 12i No. 2 1i Subspace: S 11i S 12i < No. 3 1i Subspace: S 11i S 12i. (A.1) Since the proofs for the other subspaces are similar we just prove the conclusions in No. 2 1i No.3 1i in this lemma. Invoking (11) and (13) the function δ 1i (t) can be written as the following equation for every point (e 1i e 1i ) in No. 2 i Subspace δ 1i (t) =M 1i h 1i = ω 12i1 e 1i ω 12i2 e 1i 1 = [(ξ ξ 11i ξ 12i ω 13i2 )(e 1i +ξ 1i e 1i ) 12i +(ξ 11i ω 1i2 )(e 1i +ξ 2i e 1i )]. (A.2) Since the point (e 1i e 1i ) islocatedinno.2 1i Subspace there exists S 11i =e 1i +ξ 11i e 1i S 12i =e 1i +ξ 12i e 1i <. (A.3) From (1) and (11) it is noted that ξ 11i > ξ 12i > ξ 12i < ω 1i2 <ξ 11i andξ 12i <ω 13i2 <ξ 11i. Then we can obtain that δ 1i (t) = M 1i h 1i (P 1 )>for every point (e 1i e 1i ) in No. 2 1i Subspace. Invoking (11) and (13) the function δ 1i (t) can be written as the following equation for every point (e 1i e 1i ) in No. 3 i Subspace: δ 1i (t) =M 1i h 1i =e 1i +ω 13i2 e 1i. (A.4) Since the point (e 1i e 1i ) islocatedinno.3 1i Subspace there exists S 11i =e 1i +ξ 11i e 1i S 12i =e 1i +ξ 12i e 1i. (A.5) From (1) and (11) it is noted that ξ 11i >ξ 12i >and ξ 12i < ω 13i2 <ξ 11i. Then the coefficient ω 13i2 can be expressed as ω 13i2 =ξ 12i +τ 1i (A.6) where < τ 1i < 1. Substituting (A.6) into (A.4) and considering (A.5) yield δ 1i (t) =(1 τ 1i )(e 1i +ξ 12i e 1i ) +τ 1i (e 1i +ξ 11i e 1i ). (A.7)

9 Mathematical Problems in Engineering δ e (deg) δ a (deg) δ r (deg) δ x (deg) δ y (deg) δ z (deg) Figure 5: Adaptive UAS-SMC + NDO attitude control input. Thus there exists δ 1i (t) for every point (e 1i e 1i ) in No. 3 1i Subspace. Similarly we have δ 1i (t) > for every point (e 1i e 1i ) in No. 1 1i and 2 1i Subspace. According to above discussion there exists δ 1i (t) for every (e 1i e 1i ) e 1i R e 1i R.And the function δ 1i (t) will satisfy the following equation: No. 1i Subspace: δ 1i (t) > No. 1 1i Subspace: δ 1i (t) > No. 2 1i Subspace: δ 1i (t) > No. 3 1i Subspace: δ 1i (t). (A.8) When δ 1i (t) = it is clear that the point (e 1i e 1i ) is located in No. 3 i Subspace. Substituting δ 1i (t) = into (A.7) and considering <τ 1i <1 S 11i =e 1i +ξ 11i e 1i ands 12i = e 1i +ξ 12i e 1i yield e 1i +ξ 11i e 1i = e 1i +ξ 12i e 1i =. (A.9) Considering ξ 11i =ξ 12i wehavee 1i = e 1i =.Hence if δ 1i (t) = there exists P 1 =(e 1i e 1i ) = ( ). Andwe can also obtain the similar conclusion for P 2 =(e 2i e 2i ) and δ 2i (t). B. Proof of Lemma 3 Take δ 1i (t) = M 1i h 1i (P 1 ) asanexample;wehavethe following discussions. Invoking (11) and (13) the function δ 1i (t) can be written as where δ 1i (t) = ω 1i1 e 1i ω 1i2 e 1i 1 S 11i <; S 12i < ω ω 1i1 = 11i1 S 11i <; S 12i ω 12i1 S 11i ; S 12i < 1 S 11i ; S 12i ; ω 1i2 S 11i <; S 12i < ω 11i2 S 11i <; S 12i ω 1i2 = ω 12i2 S 11i ; S 12i < ω 13i2 S 11i ; S 12i (B.1)

10 1 Mathematical Problems in Engineering ω 11i1 = [ (ξ 11i +ξ 12i )+(ω 1i2 +ω 13i2 )] ω 11i2 = [ 2ξ 11iξ 12i +(ω 1i2 ξ 11i +ω 13i2 ξ 12i )] ω 12i1 = [(ξ 11i +ξ 12i ) (ω 1i2 +ω 13i2 )] ω 12i2 = [2ξ 11iξ 12i (ω 13i2 ξ 11i +ω 1i2 ξ 12i )]. (B.2) From the definition of ω 1i1 ω 1i2 functionδ 1i is switched on switching surfaces S 11i =and S 12i =. Then the discussion for the continuity of function δ 1i can be given as follows. For every point (e 1i e 1i ) on switching surface S 11i = (1) yields e 1i = ξ 11i e 1i. (B.3) Assuming S 12i <and considering the definition of ω 1i1 ω 1i2 there exists (B.4) for S 11i = + and S 12i <: δ + 1i = ω 12i1 e 1i ω 12i2 e 1i = (ξ 11i +ξ 12i ) (ω 1i2 +ω 13i2 ) ξ 11i ξ 12i e 1i 2ξ 11iξ 12i (ω 13i2 ξ 11i +ω 1i2 ξ 12i ) ξ 11i ξ 12i e 1i. Substituting (B.3) into (B.4) yields δ + 1i =(ξ 11i ω 1i2 ) e 1i. (B.4) (B.5) Assuming S 12i <and considering the definition of ω 1i1 ω 1i2 there exists (B.6) for S 11i = <and S 12i <: δ 1i = e 1i ω 1i2 e 1i. Substituting (B.3) into (B.6) yields δ 1i =(ξ 11i ω 1i2 ) e 1i. (B.6) (B.7) Since δ + 1i = δ 1i functionδ 1i is continuous on switching surface S 11i = while S 12i <. Similarly function δ 1i is continuous on switching surface S 11i = while S 12i. Hence δ 1i is a continuous function for every point (e 1i e 1i ) on switching surface S 11i =. And we can also have the same result for every point (e 1i e 1i ) on switching surface S 12i =. From the above discussion function δ 1i (t) is continuous on switching S 11i = S 12i =. Then we can obtain that the function δ 1i (t) in Lemma 2 is a continuous function for all e 1i R.Meanwhilewecanobtainthesameconclusionfor δ 2i (t) in Lemma 2. Conflict of Interests The authors declare that there is no conflict of interests regarding the publication of this paper. Acknowledgments This work was partially supported by National Natural Science Foundation (NNSF) of China under Grants and ; Postdoctoral Fund of Jiangsu Province 14123; Pre-Research Foundation of General Equipment Department 914C33514C314. References [1] B. Jiang Z. Gao P. Shi and Y. Xu Adaptive fault-tolerant tracking control of near-space vehicle using Takagi-Sugeno fuzzy models IEEE Transactions on Fuzzy Systemsvol.18no. 5 pp [2] B.JiangY.XuP.ShiandZ.Gao Robustfault-toleranttracking control for a near-space vehicle using a sliding mode approach Proceedings of the Institution of Mechanical Engineers. Part I: Systems and Control Engineering vol.225no.8pp [3] Y. Du Q. Wu C. Jiang and J. Wen Adaptive functional link network control of near-space vehicles with dynamical uncertainties Systems Engineering and Electronics vol. 21 no. 5 pp [4]D.HouQ.WangandC.Dong Outputfeedbackdynamic surface controller design for airbreathing hypersonic flight vehicle Automatica Sinicavol.2no.2pp [5]J.YangZ.ZhaoS.LiandW.X.Zheng Nonlineardisturbance observer enhanced predictive control for airbreathing hypersonic vehicles in Proceedings of the 33rd Chinese Control Conference (CCC 14) pp Nanjing China July 214. [6] H. Sun S. Li and C. Sun Adaptive fault-tolerant controller design for airbreathing hypersonic vehicle with input saturation Systems Engineering and Electronicsvol.24no. 3 pp [7] J. Yang S. Li C. Sun and L. Guo Nonlinear-disturbanceobserver-based robust flight control for airbreathing hypersonic vehicles IEEE Transactions on Aerospace and Electronic Systemsvol.49no.2pp [8] Z.-Y. Liu H.-N. Wu and L. Guo Disturbance-observerbased guaranteed cost fuzzy control for airbreathing hypersonic vehicles in Proceedings of the 32nd Chinese Control Conference (CCC 13) pp Xi an China July 213. [9] H. Sun S. Li Y. Jun and G. Lei Nonlinear disturbance observer-based backstepping control for airbrathing hypersonic vehciles with mismatched disturbances Control Theory & Applicationsvol.8no.17pp [1] J.-T. Chang D.-R. Yu and W. Bao Characteristic analysis of unstart/restart of hypersonic inlets caused by variations of attack angle of freestream Aerospace Powervol.23 no.5pp [11] Z. Pu X. Tan G. Fan and J. Yi Uncertainty analysis and robust trajectory linearization control of a flexible air-breathing hypersonic vehicle Acta Astronauticavol.11no.1pp [12] H.LiZ.LiS.XueandD.Xu Compositecontroldesignwith sliding mode for a hypersonic aircraft in Proceedings of the 3rd

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