Adaptive sliding mode backstepping control for near space vehicles considering engine faults

Transcription

1 Journal of Systems Engineering and Electronics Vol. 9, No., April 018, pp Adaptive sliding mode backstepping control for near space vehicles considering engine faults ZHAO Jing 1,4,6,JIANGBin,XIEFei 3,4,*, GAO Zhifeng 1, and XU Yufei 5 1. College of Automation Engineering, Nanjing University of Posts and Telecommunications, Nanjing 1003, China;. College of Automation Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 10016, China; 3. School of Electrical and Automation Engineering, Nanjing Normal University, Nanjing 10046, China; 4. Jiangsu Province 3D Printing Equipment and Manufacturing Key Lab, Nanjing 1004, China; 5. Shanghai Institute of Satellite Engineering, Shanghai 0040, China; 6. Jiangsu Engineering Laboratory for Internet of Things and Intelligent Robots, Nanjing 1003, China Abstract: A fault tolerant control methodology based adaptive sliding mode (ASM) backstepping is proposed for near space vehicle (NSV) attitude control system under engine faults. The proposed scheme combined adaptive backstepping with the sliding mode control strategy could guarantee the system s stability and track desired signals under external disturbances and engine faults. Firstly, attitude mode description and the engine faulty model are given. Secondly, a nominal control law is designed. Thirdly, a sliding mode observer is given later in order to estimate both the information of engine faults and external disturbances. An adaptive sliding mode technology based on the previous nominal control law is developed via updating faulty parameters. Finally, analyze the system s fault-tolerant performance and reliability through experiment simulation, which verifies the proposed design of fault-tolerant control can tolerate engine faults, as well as the strong robustness for external disturbance. Keywords: fault tolerant control, adaptive sliding mode (ASM), engine fault, near space vehicle (NSV). DOI: /JSEE Introduction A great amount of effort has been made on the control scheme for near space vehicles (NSVs) recently, which provides valuable reliability of the flight control system, see for examples, [1 11] and the references therein. Shtessel et al. proposed sliding mode controls for NSVs [1 3], and an uncertainty modeling and fixed order controller Manuscript received November 8, 016. *Corresponding author. This work was supported by the National Natural Science Foundation of China ( ; ; ), the Natural Science Foundation of Jiangsu (BK016101), the Nanjing University of Posts and Telecommunications Science Foundation (NY14173), and the Open Program of Jiangsu Key Laboratory of 3D Printing Equipment and Manufacturing (3DL01607). were designed in [4]. A robust nonlinear control approach was presented in [5 7]. To achieve better performance of controllers for NSVs, the advantages of adaptive with robust or sliding mode control are combined [8 14]. In a flight process of NSVs, generally there are four stages, such as, the subsonic stage, transonic stage, supersonic stage and hypersonic stage. Consequently, NSV has strong coupling, complexity and serious nonlinearity. The above mentioned factors may give rise to faults, which inevitably affect flight reliability, stability and security, and consequently lead to disaster. Thus, fault tolerance control (FTC) technology for flight control system is crucial in the field of aerospace industry. Generally, fault types are classified into the following three categories [15,16]: actuator, sensor, and component fault. In [17 4], there have been some results aiming at actuator faults and structural faults for NSVs. In the existing literature, the second-order dynamic terminal sliding mode technique have been applied to study FTC approaches of NSVs, such as, Zhao et al. proposed an FTC strategy for NSV attitude system to reduce the bad effects generated by actuator faults, uncertainties and exogenous disturbances [17]. In [18], as NSVs suffer from control effector damage and actuator dynamic fault, a fault estimation and fault tolerant control scheme are developed using adaptive sliding mode (ASM). Furthermore, some important results on the active FTC approach for actuator faults were proposed in [19 ]. Alwi et al. provided some new research progress on component faults [3,4]. The issue of component faults has been investigated for aircraft dynamical systems with partial loss vertical tail in [3]. There have been rich researches on sensor faults, see for examples, [5 9] and the references therein. Lee et al. designed FTCs for sensor fault based on structured kernel principal component analysis [5]. In [6], an FTC de-

2 344 Journal of Systems Engineering and Electronics Vol. 9, No., April 018 sign subject to sensor faults has been given in the active FTC framework, specifically, by introducing a descriptor system approach, the problem of sensor fault reconstruction and compensation have been studied for a class of nonlinear systems [8]. In [30 33] different fault tolerant control schemes for structural fault were proposed. An adaptive nonsingular terminal sliding mode controller is proposed for NSVs with parameter uncertainty and external disturbance, what is more, lumped disturbance can be accommodated using an adaptive update law [34]. An adaptive sliding mode backstepping controller is designed to cope with external disturbances and parameter variations for NSVs re-entry attitude system [35]. A fault accommodation scheme is proposed for NSVs with control effector damage using both adaptive neural observer and backstepping technique [36]. An active fault tolerant control strategy is developed for NSVs under actuator fault based on Takagi-Sugeno (T-S) fuzzy model, which is designed by sliding mode observers and two fault accommodation approaches [37]. A decentralized fault-tolerant controller is designed for NSV reentry attitude dynamic to enable global stability of NSV with actuator faults and control surface damage by ASM and backstepping control [38]. The proposed robust controller of flight system with adaptive gain can guarantee accuracy tracking in the presence of nonlinearities and disturbances [39]. An adaptive neural prescribed performance control approach is proposed to enable the NSVs to track the position and attitude of the target under the influence of input nonlinearity, which consists of input saturation and dead-zone [40]. However, it is the fact that very limited existing reach results on FTCs for NSVs engine faults, have been reported in academic publications. In [41], in order to remove the effect of engine faults, a fault-tolerant controller is developed by using the indirect robust adaptive technique cooperating with the backstepping procedure and the global asymptotically attitude tracking can be achieved in the case of engine faults. Due to the inherent advance on the application of aeronautical engineering, backstepping control has been applied in flight control system [4]. Compared with existing works on faulty system of NSV, the contributions from this paper are in two-fold: (i) the unmodeled engine faults are introduced into the attitude control system of NSV; and (ii) an ASM fault-tolerant controller is proposed for engine fault cases. Section describes engine faulty dynamics of NSV and some necessary assumptions. Problem formulation is presented. In Section 3 FTC design is given for the NSV faulty model which contains exogenous disturbances and engine faults. In Section 4, simulation results and some necessary comparisons with existing work demonstrate the validity of FTC design and the superiorities contrast to some existing work. Finally, conclusion is presented in Section 5.. Modeling of NSV.1 Attitude dynamic description The attitude angle γ and attitude angular rate ω are described in the following: { J ω = ΩJω + u γ = Rω (1) where J R 3 3 denotes the inertia tensor in the form of J xx J xy J xz J = J yx J yy J yz J zx J zy J zz where ω =[p, q, r] T denotes the angular rate and p, q, r are roll velocity, pitch velocity and yaw velocity. u is the control torque generated by engine and aerodynamic surface, and it is written as u = Dδ () where D( ) R 3 6 denotes the sensitivity matrix. δ R 6 1 is the control input vector. γ = [ϕ, β, α] T where ϕ, α, β are bank angle, angle of attack and sideslip angle respectively. Ω is a skew symmetric matrix and given by 0 r q Ω = r 0 p (3) q p 0 and cos α 0 sinα R = sin α 0 cos α. (4) Let x 1 = γ and x = ω, then system (1) can be transformed as ẋ 1 = f 1 (x 1 )x ẋ = f (x 1, x )+gδ. (5) y = x 1 From (), we obtain δ = D ( )u(t) based on dynamic inverse, where the control allocation matrix D ( ) meets D( )D ( ) =I as D ( ) =D T ( )[D( )D T ( )] 1 (6) where f 1 (x 1 ) = R( ),f (x 1, x ) = J 1 ΩJω, and g = J 1 D. In the following, a nominal controller is to be designed for the NSV system (5).. Engine fault model Remark 1 As the first order actuator dynamics u = Λ(u u c ) is considered, here the actuator dynamic is assumed to be much faster than the dynamics, consequently,

3 ZHAO Jing et al.: Adaptive sliding mode backstepping control for near space vehicles considering engine faults 345 the fast actuator dynamic generally can be neglected, that is, u u c, which could permit to assume δ δ c. For an engine fault case, the disturbance torque is considered, and system (1) can be updated as { J ω = ΩJω + uc + d(t) (7) u c =(D +ΔD)δ c where ΔD is an uncertain component caused by engine faults, and it is described as follows: { =0, no engine fault ΔD (8) 0, engine fault occurs where ΔD = {d ij },i=1,, 3; j =1,, 3,...,6. (i) No engine damage fault denotes as ΔD 0,iffor i, j,s.t.d ij 0. (ii) Engine damage fault denotes as ΔD 0,if i, j, s.t. d ij 0. d(t) R 3 1 represents the external disturbance torque vector, so we can express δ c =(D +ΔD) ( )u c. Considering external disturbances and engine faults, (7) can be rewritten as { γ = Rω ω = J 1 ΩJ + J 1 (D +ΔD)δ c + J 1 (9) d(t) Thus it needs to design a controller to compensate the influences exerted by the above factors. In view of disturbance d(t), a hypothesis is given in the following. Assumption 1 d(t) is an unknown nonlinear function, which includes external disturbances and model uncertainties. We assume that there will be a continuous bounded function ς( ), thatis, d(t) <ς( ). Let x 1 = γ and x = ω, (7) can be transformed as ẋ 1 = f 1 (x 1 )x ẋ = f (x 1, x )+Gδ c + η(t). (10) y = x 1 Similarly, f 1 (x 1 )=R( ), f (x 1, x )= JΩJω η(t) =J 1 d(t) and let G = J 1 (D + D) G 0 +ΔG, whereg 0 = J 1 D, ΔG = J 1 ΔD. Remark It is noted that the research object of unmolded part ΔD is represented by the engine fault item, and it can be estimated by adaptive technology. Let G = J 1 (D +ΔD) in (7) represent the influences of engine faults on NSV system, G 0 denote the healthy item by the engine, and ΔG represent engine faults which will be estimated later. 3. ASM backstepping control design 3.1 Problem formulation In this paper, our control objective is to design an FTC scheme which can achieve the following goals: (i) the signals of the closed-loop system are bounded; (ii) the desired signals can be accurately and timely tracked under engine fault cases. First of all, a diagram of FTC system (see Fig. 1) and a hypothesis will be given for the desired signals as follows. Fig. 1 Diagram for FTC system Assumption It is assumed that both tracking signal y c and its derivative ẏ c are piecewise continuous and bounded functions in this paper. For system (3), define errors e 1 = x 1 y c, and e = x ν, and view x as a control variable for system ẋ 1 = f 1 (x 1 )x,whereν is the virtual control law to ensure e 1 0. We have the following results given in the form of Theorem 1. Theorem 1 Consider NSV attitude dynamics (3) with engine fault free cases, a nominal control scheme (11) and (1) based on backstepping is designed, such that the asymptotic output tracking of NSV attitude control system can be guaranteed, that is lim (x 1 y c )=0 t lim (x ν) =0. t Proof There will be two steps in the following to prove the theorem. Step 1 Here ν is designed to stabilize e 1 as follows: ė 1 = ẋ 1 ẏ c = f 1 (x 1 )x ẏ c = f 1 (x 1 )(ν + e ) ẏ c. Consider a Lyapunov function in the form of V 1 = 1 et 1 e 1. The time derivative of V 1 and we obtain V 1 = e T 1 ė1 = e T 1 [f 1(x 1 )(ν + e ) ẏ c ]. An appropriate virtual control ν can be designed as follows: ν = f 1 1 (x 1)( k 1 e 1 + ẏ c ). (11)

4 346 Journal of Systems Engineering and Electronics Vol. 9, No., April 018 One obtains V 1 = e T 1 ė1 = e T 1 f 1(x 1 )e k 1 e T 1 e 1. Step The global control law δ will be designed in this step, which makes errors e 1 0, e 0. Another Lyapunov function is chosen as V = V et e. Then, we can obtain the derivative of V as V = V 1 + e T ė = k 1 e T 1 e 1 + e T 1 f 1(x 1 )e + e T ė = k 1 e T 1 e 1 + e T 1 f 1(x 1 )e + e T [f (x 1, x )+ gδ ν] = k 1 e T 1 e 1 + e T [f 1 (x 1 )e 1 + f (x 1, x )+ gδ ν ẋ 1 ν ẏ c ]. x 1 y c In the following, we will design an appropriate controller δ, which can remove some terms with e 1, e, x 1, x, δ = g 1 [ k e f 1 (x 1 )e 1 f (x 1, x )+ ν ẋ 1 + ν ẏ c ] (1) x 1 y c where k > 0, so one obtains V = k 1 e T 1 e 1 k e T e < 0 The proof of Theorem 1 has been completed. Remark 3 Note that g 1 in (1) cannot be always ensured, so g 1 is replaced by the pseudo-inverse of g # = g T [gg T ] 1. The controller (1) can be updated based on Remark 1 and Assumption, δ c = g T (gg T ) 1 [ k e f 1 (x 1 )e 1 f (x 1, x )+ 3. Fault tolerant control design ν x 1 ẋ 1 + ν y c ẏ c ]. (13) The control objective is to design the actuator deflection command δ c, which can make the output vector y tracks a command signal y c asymptotically under external disturbances and engine faults. Similarly, an ideal controller is designed for faulty system (7) as follows: δ cf =(G 0 +ΔG) 1 [ k e f 1 (x 1 )e 1 η f (x 1, x )+ ν x 1 ẋ 1 + ν y c ẏ c ] (14) where δ c denotes the FTC input for engine fault cases. Since ΔG, η are unknown in (1), so in the next, an adaptive algorithm will be designed to obtain estimated values of ΔG, andη, respectively. In the following, a dynamic sliding mode observer will be designed in order to obtain the estimation values ΔĜ, η Adaptive sliding observer design In order to estimate some parameters of controller (14), we need to design an adaptive observer to estimate both of the information for external disturbance and engine fault ΔĜ, η respectively. The adaptive observer has the form of x 1 = f 1 (x 1 ) x x x = f (x 1, x )+(G 0 +ΔĜ)δ c + η(x,t). (15) ŷ = x 1 Then the error system is obtained from (10) subtracting (15) z 1 = x 1 x 1 ż 1 = ẋ 1 x 1 = f 1 (x 1 )(x x ). The corresponding results are given in the form of Theorem. Theorem Under the dynamic sliding mode observer design (15), the adaptive laws (18), (19) for estimating ΔG and η can make system stable and meanwhile such that z 1 0. Proof A dynamic sliding mode surface is designed as follows: s = ż 1. (16) An appropriate Lyapunov function is chosen as in the form of V 3 = 1 [st s +Δ G T Γ 1 1 Δ G + η T Γ 1 η] (17) where Γ 1 and Γ are positive definite adaptive gain matrices, and η = η η, andletδ G =ΔG ΔĜ be the estimated errors, where η and ΔĜ are the estimations of η, ΔG respectively. Then the derivative of Lyapnunov of V 3 is V 3 = s T ṡ Δ G T Γ 1 1 Δ Ĝ η T Γ 1 η = s T f 1 (x 1 )(ẋ x ) Δ G T Γ 1 1 Δ Ĝ η T Γ 1 η = s T f 1 (x 1 )[(ΔG ΔĜ)δ c + η η] Δ G T Γ 1 1 Δ Ĝ η T Γ 1 η = s T f 1 (x 1 )[Δ Gδ c + η] η T Γ 1 η. Choose appropriate estimated algorithms for ΔG and η as follows: η = Γ f 1 (x 1 )s = Γ f 1 (x 1 )ż 1 (18)

5 ZHAO Jing et al.: Adaptive sliding mode backstepping control for near space vehicles considering engine faults 347 Δĝ ij =Proj [ζi,1][1 sign(δ cfi f 1 ij s i)] (19) where Proj [ζi,1] denotes the projection operator, which projects the estimate Δĝ ij to the interval [ζ i, 1], and0 ζ i < 1, sign( ) denotes the sign function. Remark 4 The estimation algorithm of engine fault (18) owns the following property referred to [14] as follows: (δ cfi f 1 ij s i)(δg ij Δĝ ij ) 0. (0) On the basis of Remark 4, the derivative of Lyapunov of V 3 further is rewritten as follows: V 3 = s T f 1 (x 1 )[Δ Gδ c + η] η T Γ 1 η = s T f 1 (x 1 )Δ Gδ c + η T [f(x 1 )s Γ 1 η] 0. So far, the proof of Theorem has been completed. 3.. Adaptive backstepping FTC design From the above analysis, some results can be obtained in the form of Theorem 3. Theorem 3 The backstepping controllers (11) and (1) updated by adaptive algorithms (18) and (19), as applied to the engine faulty model system subject to (9), could guarantee the asymptotic output tracking performance and also the boundedness of all the closed-loop signals, that is, lim t (x 1 y c ) 0. Proof follows: δ cf =(G 0 +ΔĜ) 1 [ k e f 1 (x 1 )e 1 η f (x 1, x )+ ν x 1 ẋ 1 + ν y c ẏ c ] (1) Consider an appropriate Lyapunov function as V 4 = V 1 + V et e. () Then the derivate of V 4 is as follows: V 4 = V 1 + V 3 + e T ė = k 1 e T 1 e 1 + e T 1 f 1(x 1 )e + e T [f (x 1, x )+ (G 0 +ΔG)δ c + η ν]+ s T f(x 1 )[Δ Gδ c + η] η T Γ 1 η. Based on (11) and adaptive laws (18) and (19), one obtains V 4 = V 1 + V 3 + e T ė k 1 e T 1 e 1 + e T 1 f 1(x 1 )e + e T [f (x 1, x )+ (G 0 +ΔĜ)δ c + η ν] = k 1 e T 1 e 1 + e T [f 1 (x 1 )e 1 + f (x 1, x )+ (G 0 +ΔĜ)δ c + η ν ẋ 1 ν ẏ c ]. x 1 y c One obtains via further calculating V 4 k 1 e T 1 e 1 k e T e where k 1 > 0, and k > 0, one obtains V 4 < 0. So far, the proof of theorem 3 has been completed. Remark 5 Due to [(G 0 +ΔĜ)T ] 1 in (1) cannot be always ensured, let [(G 0 +ΔĜ)T ] 1 be replaced by its pseudo-inverse (G 0 +ΔĜ)T [(G 0 +ΔĜ)(G0+ΔĜ)T ] 1. Then, (1) can be rewritten as δ c =(G 0 +ΔĜ)T [(G 0 +ΔĜ)(G 0 +ΔĜ)T ] 1 [ k e f 1 (x 1 )e 1 η f (x 1, x )+ ν ẋ 1 + ν ẏ c ]. (3) x 1 y c Remark 6 The effect of engine faults for NSV flight control system is slowly changing, and the main effect objects are height and velocity during the cruise phase. The effect of attitude control system can be compensated by the rudder control allocation. Thus the proposed method can guarantee attitude system s stability. 4. Example In the following, the proposed method in this paper will be verified by simulation experiments, we choose the initial state: γ(0) = [0, 0, 0] T, ω(0) = [0., 0, 0.4] T,and tracking signals, γ d =[1, 0, ] T, ω d =[0, 0, 0] T, respectively J = The disturbance d(t) with upper bound d(t) =10 is denoted by cos0.05t d(t) = cos0.05t cos0.05t EnginedamagefactorΔD is given as follows: { diag{0, 0, 0}D, t < 15s ΔD = diag{0.1, 0.1, 0.1}D, 15s <t<100s and k 1 = k =0.3, Γ 1 = Γ =diag{0.5}. To verify the effectiveness of the proposed FTC approach, we consider the following different cases during simulation experiments to observe the responses of control torques u = [u 1,u,u 3 ] T, attitude rates w = [w 1,w,w 3 ] T and attitude angles x =[x 11,x 1,x 13 ] T. Case A No engine fault In this case, ΔD =0, that is, there is no engine fault. The corresponding simulation results using the method

6 348 Journal of Systems Engineering and Electronics Vol. 9, No., April 018 proposed nominal control law (1) in this paper are depicted in Fig. (a) Fig. 8(a). From Fig. (a) Fig. 4(a), it can be seen that, control torques u 1,u,u 3 can be stable quickly in engine fault free cases. Similarly, the responses of ω 1,ω,ω 3 and x 11,x 1,x 13 can be seen respectively from Fig. 5(a) Fig. 7(a), which show that tracking signal is tracked accurately and quickly. Case B Engine fault without FTC As the engine fault occurs, that is, ΔD 0,inorder to compare the proposed method, we experiment without fault tolerance first, and the corresponding simulation results of the nominal control law (1) are demonstrated in Fig. (b) Fig. 8(b), which show that control torques responses of u 1,u begin to wave after 15 s. While the influence of the engine fault on u 3 is not so obvious from Fig. 4(b). Fig. 5(b) Fig. 8(b) show the responses of attitude rates ω 1,ω,ω 3 and the states of x 11,x 1,x 13. It can be seen that system performance requirements cannot be met with the nominal controller law, due to the angle velocities ω 1,ω,andx 11 cannot track the desired signals once the engine fault occurs. Fig. Responses of u 1 under cases A, B, C Fig. 3 Responses of u under cases A, B, C Fig. 4 Responses of u 3 under cases A, B, C

7 ZHAO Jing et al.: Adaptive sliding mode backstepping control for near space vehicles considering engine faults 349 Fig. 5 Responses of ω 1 under cases A, B, C Fig. 6 Responses of ω under cases A, B, C Fig. 7 Responses of ω 3 under cases A, B, C Fig. 8 Responses of x under cases A, B, C

8 350 Journal of Systems Engineering and Electronics Vol. 9, No., April 018 Case C Engine fault with FTC Similarly, as the engine fault occurs, that is, ΔD 0, and the corresponding simulation results are expressed as Fig. (c) Fig. 8(c). Compared to the designed control law (1), it can be easily seen that the tracking precision of control torque u can be markedly improved, especially for u 1, u from Fig. 6(b) Fig. 7(b). ω and ω 3 can also track the desired signals within 0 s while ω 1 has a slight fluctuation near the tracking target. Fig. 8(b), shows attitude angles responses of NSV s nonlinear dynamics under engine faults but without FTC technique. Obviously, the responses of x 11 begin to show a big fluctuation near the tracking signal. While from Fig. 8(c), we can see that the proposed fault tolerant control method in this paper could make the system accurately track the desired signals. As the engine fault occurs, the proposed FTC scheme can compensate the engine fault within effective time, and consequently keep the flight control system stable. In summary, the simulation results can prove that the proposed scheme has better tolerance to engine faults from the simulation results. 5. Conclusions In this paper, an FTC technique based on ASM backstepping is proposed for the attitude control systems of NSV considering engine faults and external disturbances. The proposed scheme combining adaptive backstepping with the sliding mode control strategy could guarantee the stability of the system and track the desired signals under external disturbances and engine faults. Firstly, attitude mode description and the engine faulty model are given. Secondly, a nominal control law is designed. Thirdly, a sliding mode observer is given to estimate both the information of engine faults and external disturbances. 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