Anti-disturbance control of hypersonic flight vehicles with input saturation using disturbance observer

Transcription

1 . RESEARCH PAPER. SCIENCE CHINA Inormation Sciences Special Focus on Advanced Nonlinear Control o Hypersonic Flight Vehicles July 215, Vol :1 722:12 doi: 1.17/s Anti-disturbance control o hypersonic light vehicles with input saturation using disturbance observer CHEN Mou 1 *,RENBeiBei 2,WUQinXian 1 & JIANG ChangSheng 1 1 College o Automation Engineering, Nanjing University o Aeronautics and Astronautics, Nanjing 2116, China; 2 Department o Mechanical Engineering, Texas Tech University, Lubbock , USA Received November 2, 214; accepted February 14, 215; published online May 18, 215 Abstract This paper proposes an anti-disturbance control scheme or the near space vehicle (NSV) based on terminal sliding mode (TSM) technique and disturbance observer method. To tackle the system uncertainty and the time-varying unknown external disturbance o the NSV, a disturbance observer based on TSM technique is designed which can render the disturbance estimate error convergent in inite time. Furthermore, an auxiliary design system is introduced to analyze the input saturation eect. Based on the developed disturbance observer and the auxiliary design system, an anti-disturbance attitude control scheme is developed or the NSV using the TSM technique to speed up the convergence o all signals in closed-loop system. For the closed-loop system, the stability is rigorously proved by using the Lyapunov method and we guarantee the inite time convergence o all closed-loop system signals in the presence o the integrated aection o the system uncertainty, the input saturation, and the unknown external disturbance. Simulation study results are given to show the eectiveness o the developed TSM anti-disturbance attitude control scheme using the disturbance observer and the auxiliary system or the NSV. Keywords observer near space vehicles, anti-disturbance control, sliding mode control, input saturation, disturbance Citation Chen M, Ren B B, Wu Q X, et al. Anti-disturbance control o hypersonic light vehicles with input saturation using disturbance observer. Sci China In Sci, 215, 58: 722(12), doi: 1.17/s Introduction Nowadays, the hypersonic light vehicle (HFV) has attracted much attention due to its wide application in both civilian and military areas [1]. Indeed, there are a number o robust light control schemes developed or the HFV [2,3]. In [4], a direct neural discrete control scheme was proposed or the HFV. Fuzzy tracking control scheme was studied or the HFV via Takagi-Sugeno (T-S) model in [5]. In [6], an adaptive neural control scheme was proposed based on high gain observer or HFVs. The online support vector regression compensated nonlinear generalized predictive control was developed or the HFV in [7]. The near space vehicle (NSV) as one o the most important HFVs has received increasing interest in recent years due to working in the near space and perorming various tasks. As a kind o HFV, the NSV processes a wide range o light envelopes and the light environment is sharply changeable with time variation. Consequently, it is necessary to develop eicient light control schemes to enhance the control robustness. As is well known, the NSV has important eatures, such as high control channel coupling and * Corresponding author ( chenmou@nuaa.edu.cn) c Science China Press and Springer-Verlag Berlin Heidelberg 215 ino.scichina.com link.springer.com

2 Chen M, et al. Sci China In Sci July 215 Vol :2 strong nonlinearity, which will urther increase the design diiculty o the robust light control scheme by considering the unknown external disturbance. Thus, the robust light control design o the NSV is a signiicant challenge in the control area. With the NSV being a multiple-input multiple-output (MIMO) nonlinear system subject to external disturbances and large uncertainties, it is natural to develop eicient robust light control schemes or the NSV [8 13]. For an NSV, an adaptive light control scheme was developed using unctional link neural network in [14]. In [15], a robust control scheme was studied or the NSV with input backlash-like hysteresis. Using T-S uzzy technique, a ault-tolerant light control scheme was developed or the NSV in [16]. In [17], a ault tolerant-control scheme was studied or the NSV based on backstepping method. By reviewing these research results, it is clear that the robust light control design needs to be urther investigated or the NSV subjected to unknown external disturbance and input saturation. To eiciently handle the unknown disturbance and ully use the dynamic inormation o disturbance, the disturbance observer can be introduced to design the light control scheme o the NSV. In the past several decades, the disturbance-observer based control (DOBC) schemes have been extensively studied [18 22]. An adaptive tracking control scheme was developed using uzzy logic system or a MIMO nonlinear systems with disturbance observer in [23]. In [24], an adaptive control scheme was proposed using neural network and disturbance observer or the nonaine nonlinear system. For the bank-to-turn missile, a robust autopilot was designed based on disturbance observer in [25]. In [26], a disturbance suppression control scheme was proposed or nonlinear systems using DOBC method. During the past years, some disturbance observer-based robust light control schemes have been developed or the NSV to improve the disturbance rejection ability. Robust optimal predictive control was developed or the NSV based on unctional link network disturbance observer in [27]. In [28], a robust disturbance observer based-attitude control was proposed or the NSV with external disturbance. Sliding mode disturbance observer (SMDO) has been extensively studied as a class o nonlinear disturbance observers. However, the convergence speed o the disturbance estimate error needs to be urther improved to meet the control requirement o the NSV. In this paper, recalling the terminal sliding mode disturbance observer (TSMDO) developed or the single-input and single-output system [29], the disturbance observer with inite time convergence property will be developed or the NSV. On the other hand, there exists the input saturation due to only limited orces and moments provided by the equipped actuators o the NSV [3 35]. To ensure satisactory control perormance o the NSV, the input saturation needs to be considered. In [36], an adaptive tracking control was studied using neural network or a class o stochastic nonlinear systems subject to unknown input saturation. Tracking control scheme was studied or nonlinear systems with bounded controls and control rates in [37]. In [38], a guaranteed transient perormance-based control was proposed or the NSV with input saturation. Although some control schemes have been designed to control an NSV with input saturation, the aster control speed is desired or the NSV to complete the expected control objective. In this setting, the terminal sliding mode (TSM) technique can be employed to develop the light control scheme o the NSV. Up to now, the TSM control schemes have been studied or various aircrats. In [39], the six-degree-oreedom robust and adaptive TSM control scheme was studied or the ormation lying o spacecrats. Second-order TSM control scheme was proposed or the cruising light o a hypersonic vehicle based on the SMDO in [4]. In [41], a second-order nonsingular TSM control was developed or a reusable launch vehicle. To speed the control convergence, the TSM control strategy should be urther investigated or the NSV in the presence o input saturation, system uncertainty, and unknown disturbance. According to the above discussion, the disturbance observer-based TSM control to ollow the desired attitude command signals o the NSV subject to the unknown disturbance and the input saturation. The layout o the paper is as ollows. The problem description o the TSM attitude control or the NSV is given in Section 2. Section 3 describes the tracking control design or a class o MIMO nonlinear systems using the TSM technique and the disturbance observer. The TSM attitude control schemes are presented or the NSV using the developed disturbance observer-based TSM control scheme in Section 4. Simulation study results are provided to show the eectiveness o the proposed SMDO-based attitude control approach in Section 5. Some conclusion remarks are addressed in Section 6.

3 Chen M, et al. Sci China In Sci July 215 Vol :3 2 Problem description To develop the anti-disturbance attitude control scheme using the TSM technique or the NSV, the attitude motion model is achieved according to the six-degrees-o-reedom and twelve-state kinematic equations, which is expressed as [7] Ω =F s (Ω) + G s (Ω)ω + D s (Ω,d s ), y s =Ω, (1) ω = F (ω)+g (ω)m c + D (ω, d ), y = ω, (2) where Ω = [α, β, μ] T is a attitude angle vector consisting o angle o attack, sideslip angle, and light-path roll angle, respectively; ω =[P, Q, R] T is the ast-loop state vector including three body-axis angular velocities; and M c =[l ctrl,m ctrl,n ctrl ] T is a control moment vector o the NSV. The expressions o F s, G s, F,andG are omitted here and can be ound in [7]. D s (Ω,d s )=ΔF s (Ω) + d s (t), D (ω, d )= ΔF (ω)+d (t), where ΔF s (Ω) and ΔF (ω) denote the system uncertainties, and d s (t) andd (t) arethe unknown external disturbances. In this paper, the attitude angle vector Ω and the body-axis angular velocity vector ω are assumed to be measurable. In general, there exist the constraints o rated angular velocities in light process o the NSV, which lead to the bounded angular velocities. On the other hand, the bounded moments are generated by the bounded surace delection angles. Thus, the angular velocity vector ω and the control moment vector M c satisy the nonsymmetric saturation constraints. To easily describe these constraints, the angular velocity vector ω and the control moment vector M c are uniormly replaced by a vector l =[l 1,l 2,l 3 ] T. Deine l c =sat(l) = [sat(l 1 ), sat(l 2 ), sat(l 3 )] T and the saturation operation sat(l i ) is deined as l ri max, i l i >l ri max, sat(l i )= l i, i l li max l i l ri max, (3) l li max, i l i <l li max, where l =[l 1,l 2,l 3 ] T is the input o the saturation operation. l ri max > andl li max <,, 2, 3are the known saturation levels o the angular velocity and the control moment. To design the TSM control or the uncertain attitude motion dynamics (1) and (2) o the NSV, we give the ollowing lemma and assumptions: Lemma 1 ([29,42]). For a continuous and positive unction V (t), i the ollowing condition is held: Then, V (t) converges in inite time t s and t s is decided by V (t)+αv (t)+λv γ (t), t >t. (4) 1 t s t + α(1 + γ) ln αv 1 γ (t )+λ, (5) λ where α>, λ>, and <γ<1. Assumption 1 ([29,42]). For the attitude motion dynamics (1) and (2) o the NSV, matrices G s and G are invertible. Assumption 2. For the disturbances D s and D o the NSV, there exist positive constants λ si and λ i such that λ si > D si and λ i > D i,, 2, 3. In this paper, we design the TSM attitude control scheme to ollow a desired attitude signal o the NSV with input saturation and external disturbance. Furthermore, all closed-loop system signals are convergent in inite time under the developed TSM attitude control. 3 Anti-disturbance control o MIMO uncertain nonlinear systems According to (1) and (2), the attitude motion dynamics o the NSV can be described by an uncertain MIMO nonlinear system. To acilitate the design o the TSM attitude control scheme, we irst consider

4 Chen M, et al. Sci China In Sci July 215 Vol :4 the TSM control o a general MIMO nonlinear system, which is written as ẋ = F (x)+g(x)u + D(x, t), y = x, (6) where x R n 1 is the system measurable state, y R n 1 is the system output, and u R n 1 is the control input vector. F (x) R n 1 and G(x) R n n are the known vector and matrix o system unction and control gain, respectively. D(x, t) = ΔF (x) + d(t) is a compound disturbance which includes the system uncertainty ΔF (x) R n 1 and the disturbance d(t) R n 1. Usually, there may exist the input saturation or the uncertain MIMO nonlinear system (6). We assume that the control input u =[u 1,u 2,...,u n ] T satisies nonsymmetric saturation. That is, the control signal u(t) =sat(v) = [sat(v 1 ), sat(v 2 ),...,sat(v n )] T is constrained by v ri max, i v i >v ri max, sat(v i )= v i, i v li max v i v ri max, (7) v li max, i v i <v li max, where v =[v 1,v 2,...,v n ] T is the input o the saturation operation. v ri max > andv li max <, i = 1, 2,...,n are the known saturation levels o the control input. Deine Δu = u v. Suppose that the dierence Δu between the desired control input v and the actual control input sat(u) satisies the ollowing assumption: Assumption 3. where θ is a known constant. For the Δu, it is assumed to satisy the ollowing condition: Δu θ, (8) For a given system output y d, the developed TSM control scheme should render all closed-loop signals convergent in inite time. For the studied MIMO nonlinear system (6), the matrix G(x) is assumed to be invertible in the anti-disturbance control design. Remark 1. From the view o a practical control system, the dierence Δu between the desired control input u and the actual control input sat(u) cannot be large. The reason is that the system controllability should be satisied when control input saturation occurs. For example, the provided control moment M c is bounded due to the bounded surace delection angle or the NSV. However, the NSV is controllable under the bounded surace delection angles. Thus, we can conclude that the dierence Δu is bounded. To satisy Assumption 3, the parameter θ can be large. 3.1 Design o disturbance observer with inite time convergence property To design a TSMDO or the disturbance D o the uncertain MIMO nonlinear system (6), an auxiliary variable is deined as [29] σ = z x, (9) where z is given by ż = Kσ Λsign(σ) Γσ p/q F (x) sign(σ)+g(x)u, (1) where q and p are odd positive integers with p <q. K = diag{k i } n n, Λ = diag{λ i } n n,γ= diag{τ i } n n, k i >, λ i > andτ i > are design parameters and λ i > D i, σ p/q =[σ p/q 1,σ p/q 2,..., σn p/q ] T,sign(σ) = [sign(σ 1 ), sign(σ 2 ),...,sign(σ n )] T,and F (x) = diag{ F i (x) } n n. The disturbance estimate ˆD o the TSMDO is decided by ˆD = Kσ Λsign(σ) Γσ p/q F (x) sign(σ) F (x). (11) Dierentiating (9), and considering (6) and (1) yields σ =ż ẋ = Kσ Λsign(σ) Γσ p/q F (x) sign(σ) F (x) D. (12) The above design o the TSMDO can be included in the ollowing theorem.

5 Chen M, et al. Sci China In Sci July 215 Vol :5 Theorem 1. For an MIMO nonlinear system described by (6), the TSMDO is designed in accordance with (9) (11). Then, the inite time convergence o the estimate error can be guaranteed or the developed TSMDO. Proo. Consider the ollowing Lyapunov unction candidate V o = 1 2 σt σ. (13) Considering (12), we have V o = σ i σ i = [σ i ( k i σ i λ i sign(σ i ) τ i σ p/q i F i (x) sign(σ i ) F i (x) D i )] k 2 i σi 2 λ i σ i n k 2 i σi 2 τ i σ (p+q)/q i τ i σ (p+q)/q i F i (x) σ i + F i (x) σ i + σ i D i 2kV o 2 (p+q)/2q τv (p+q)/2q o, (14) where k = min(k i )andτ = min(τ i ). Deine D = D ˆD. Considering (6) and (9) (11), we obtain D = D ˆD = D + Kσ +Λsign(σ)+Γσ p/q + F (x) sign(σ)+f(x) = ẋ F (x) G(x)u + Kσ +Λsign(σ)+Γσ p/q + F (x) sign(σ)+f(x) = ẋ + Kσ +Λsign(σ)+Γσ p/q + F (x) sign(σ) G(x)u = ẋ ż = σ. (15) Based on Lemma 1 and (14), we can guarantee the auxiliary variable σ converge to zero in inite time. Since the auxiliary variable σ converges to zero in inite time, we know that the derivative o σ can converge to zero in inite time. Thus, the inite time convergence o the disturbance estimate error D o the designed TSMDO can also be guaranteed according to (15). This concludes the proo. 3.2 Design o tracking control based on TSM technique In this subsection, the TSM control scheme will be studied or the uncertain MIMO system (6). To handle the input saturation o the uncertain MIMO nonlinear system (6), the ollowing auxiliary system is constructed to compensate or the eect o the input saturation with the same order as the MIMO nonlinear system [43]: ξ = Mξ Wξ p1/q1 θgsign(ξ)+g(x)δu, (16) where q 1 and p 1 are odd positive integers with p 1 < q 1. ξ R n is the state o the auxiliary system. M = diag{m i } n n, W = diag{w i } n n, m i > andw i > are design parameters. sign(ξ) = [sign(ξ 1 ), sign(ξ 2 ),...,sign(ξ n )] T, g = diag{ g i (x) } n n, ξ p1/q1 =[ξ p1/q1 1,ξ p1/q1 2,...,ξn p1/q1 ] T, g i (x) isthe ith row o the control gain matrix G(x). Choose the Lyapunov unction candidate as Invoking (16), we have V a = ξ T Mξ w i ξ (p1+q1)/q1 i n m 2 i ξi 2 w i ξ (p1+q1)/q1 i V a = 1 2 ξt ξ. (17) θξ T gsign(ξ)+ξ T G(x)Δu θ g i (x) ξ i + ξ T G(x)Δu

6 Chen M, et al. Sci China In Sci July 215 Vol :6 2mV a 2 (p1+q1)/2q1 wv (p1+q1)/2q1 a, (18) where m = min(m i )andw = min(w i ). To design the TSM control scheme or the studied MIMO nonlinear system (6), we deine the sliding surace as s = z y d ξ. (19) In accordance with (6), (9), and Δu = u v, the derivative o s is ṡ = ẋ + σ ẏ d ξ = F (x)+g(x)u + D(x, t) ẏ d + Mξ + Wξ p1/q1 + θgsign(ξ) G(x)Δu + σ = F (x)+g(x)(v +Δu)+D(x, t) ẏ d + Mξ + Wξ p1/q1 + θgsign(ξ) G(x)Δu + σ = F (x)+g(x)v + D(x, t) ẏ d + Mξ + Wξ p1/q1 + θgsign(ξ)+ σ. (2) Using the designed TSMDO and the auxiliary system, the TSM tracking control scheme is proposed as v = G 1 (x)(f (x) ẏ d + ˆD + Mξ + Wξ p1/q1 + θgsign(ξ)+bs + Ls p2/q2 ), (21) where q 2 and p 2 are odd positive integers with p 2 <q 2. B = diag{b i } n n, L = diag{l i } n n, b i > and l i > are design parameters and s p2/q2 =[s p2/q2 1,s p2/q2 2,...,s p2/q2 n ] T. The above development o the TSM control using the TSMDO or the MIMO nonlinear system (6) is summarized as ollows: Theorem 2. Considering an uncertain MIMO nonlinear system (6) subject to the disturbance and the input saturation (7), the TSMDO is designed as (9) (11). Using the developed disturbance observer-based TSM tracking control scheme (21), the inite time convergence o all closed-loop signals are guaranteed. Proo. Considering (2) and (21), we have ṡ = Bs Ls p2/q2 + D ˆD + σ = Bs Ls p2/q2 + D + σ. (22) Invoking (15), we obtain ṡ = Bs Ls p2/q2. (23) Consider the Lyapunov unction candidate V c = 1 2 st s. (24) Invoking (23) and (24), the time derivative o V is V c = b i s 2 i l i s (p2+q2)/q2 2bV c l2 (p2+q2)/2q2 V c (p2+q2)/2q2, (25) where b = min(b i )andl = min(l i ). According to Lemma 1 and (25), we obtain s in inite time, which means z y d in inite time by considering ξ in inite time. From the deinition o z, weknowσ + x y d. Since σ in inite time, we can obtain x y d in inite time. Namely, the tracking error is convergent in inite time under the designed TSM control scheme. For the whole system, including the disturbance observer and the auxiliary system, the Lyapunov unction candidate is chosen as V = V o + V a + V c. (26) Considering (14), (18), and (25), we have where V = V o + V a + V c 2δ 1 V δ 2 2 (p+q)/2q V (p+q)/2q, (27) δ 1 = min(k, m, r), δ 2 = min(τ i,w i,l i ), p + q 2q ( p + q =min, p 1 + q 1, p ) 2 + q 2. 2q 2q 1 2q 2 From (27) and Lemma 1, we know that the inite time convergence o all closed-loop system signals can be guaranteed. This concludes the proo.

7 Chen M, et al. Sci China In Sci July 215 Vol :7 Remark 2. To estimate the unknown disturbance in inite time, the TSMDO is designed or an uncertain MIMO nonlinear system in this paper. From (11), we can see that the disturbance estimate involves the sign unction. To obtain the smooth disturbance estimate, an extra low-pass ilter can be introduced in the designed disturbance observer as the same as the developed SMDO in [44]. On the other hand, the estimated output o TSMDO is employed to design TSM tracking control (21). Thus, the developed TSM tracking control based on TSMDO has a good disturbance reject ability or the uncertain MIMO nonlinear system (6). 4 Anti-disturbance attitude control design In this section, the attitude tracking control schemes will be developed or the NSV using the designed TSM tracking control approach in Section Terminal sliding mode control design or attitude angle loop According to (9) and (1), or the attitude angle dynamic (1) o the NSV, we have where z s is designed as σ s = z s Ω, (28) ż s = K s σ s Λ s sign(σ s ) Γ s σ ps/qs s F s (Ω) sign(σ s )+G s (Ω)ω, (29) where q s and p s are odd positive integers with p s <q s. K s = diag{k si } 3 3,Λ s = diag{λ si } 3 3, Γ s = diag{τ si } 3 3, k si >, λ si > andτ si >,i = 1, 2, 3 are design parameters. Here, λ si > D si, σs ps/qs =[σ ps/qs s1,σ ps/qs s2,σ ps/qs s3 ] T,sign(σ s ) = [sign(σ s1 ), sign(σ s2 ), sign(σ s3 )] T,and F s (Ω) = diag{ F s1 (Ω), F s2 (Ω), F s3 (Ω) }. In accordance with (11), the TSMDO o the attitude angle loop is proposed as ˆD s = K s σ s Λ s sign(σ s ) Γ s σ ps/qs s F s (Ω) sign(σ s ) F s (Ω). (3) To handle the constraint o rated angular velocity in light process, the ollowing auxiliary system is constructed according to (16): ξ s = M s ξ s W s ξ ps1/qs1 s θ s g s sign(ξ s )+G s (Ω)Δω, (31) where q s1 and p s1 are odd positive integers with p s1 <q s1. ξ s R 3 is the state o the auxiliary system. M s = diag{m si } 3 3, W s = diag{w si } 3 3, m si > andw si > are design parameters. sign(ξ s )= [sign(ξ s1 ), sign(ξ s2 ), sign(ξ s3 )] T, g s = diag{ g si (x) } 3 3, ξs ps1/qs1 =[ξ ps1/qs1 s1,ξ ps1/qs1 s2,ξ ps1/qs1 s3 ] T, g si is the ith row o the control gain matrix G s (Ω), Δω = ω v ω, Δω θ s,andv ω is a designed command input. To design TSM controller o the attitude angle dynamic (1), the sliding surace is designed as s s = z s Ω d ξ s, (32) where Ω d is the desired signal o the attitude angle Ω. Invoking (21), the TSM control law o the attitude angle dynamic (1) is proposed as v ω = G 1 s (Ω)(F s(ω) Ω d + ˆD s + M s ξ s + W s ξs ps1/qs1 + θ s g s sign(ξ s )+B s s s + L s s ps2/qs2 s ), (33) where q s2 and p s2 are odd positive integers with p s2 <q s2. B s = diag{b si } 3 3, L s = diag{l si } 3 3, b si > and l si >,, 2, 3 are design parameters, and s ps2/qs2 s =[s ps2/qs2 s1,s ps2/qs2 s2,s ps2/qs2 s3 ] T.

8 Chen M, et al. Sci China In Sci July 215 Vol :8 4.2 Terminal sliding mode control design or attitude angular velocity loop For the attitude angular velocity dynamic (2) o the NSV, according to (9) and (1), we have where z is designed as σ = z ω, (34) ż = K σ Λ sign(σ ) Γ σ p /q F (ω) sign(σ )+G (ω)m c, (35) where q and p are odd positive integers with p <q. K = diag{k i } 3 3,Λ = diag{λ i } 3 3, Γ = diag{τ i } 3 3, k i >, λ i > andτ i >,i =1, 2, 3 are design parameters and λ i > D i, σ p /q = [σ p /q 1,σ p /q 2,σ p /q 3 ] T, sign(σ ) = [sign(σ 1 ), sign(σ 2 ), sign(σ 3 )] T, and F (ω) = diag{ F 1 (ω), F 2 (ω), F 3 (ω) }. In accordance with (11), the TSMDO o the attitude angle velocity loop is proposed as ˆD = K σ Λ sign(σ ) Γ σ p /q F (ω) sign(σ ) F (ω). (36) To handle the constraint o the control moment, the ollowing auxiliary system is constructed according to (16): ξ = M ξ W ξ p 1/q 1 θ g sign(ξ )+G (ω)δm c, (37) where q 1 and p 1 are odd positive integers with p 1 <q 1. ξ R 3 is the state o the auxiliary system. M = diag{m i } 3 3, W = diag{w i } 3 3, m i > andw i > are design parameters. sign(ξ )= [sign(ξ 1 ), sign(ξ 2 ), sign(ξ 3 )] T, g = diag{ g i (x) } 3 3, ξ p 1/q 1 =[ξ p 1/q 1 1,ξ p 1/q 1 2 the ith row o the control gain matrix G (ω), ΔM c = M c v M, ΔM c θ,andv M command input. To design TSM controller o the attitude angle dynamic (1), the sliding surace is given by,ξ p 1/q 1 3 ] T, g i is is a designed s = z ω d ξ, (38) where ω d is the desired signal o the attitude angle ω. Invoking (21), the TSM control law o the attitude angle dynamic (2) is proposed as v M = G 1 (Ω)(F (Ω) Ω d + ˆD + M ξ + W ξ p 1/q 1 + θ g sign(ξ )+B s + L s p 2/q 2 ), (39) where q 2 and p 2 are odd positive integers with p 2 <q 2. B = diag{b i } 3 3, L = diag{l i } 3 3, b i > andl i >, i =1, 2, 3 are design parameters, and s p 2/q 2 Remark 3. =[s p 2/q 2 1,s p 2/q 2 2,s p 2/q 2 3 ] T. From (3), (31), (36), and (37), we note that the disturbance upper boundaries λ si and λ i and the upper boundaries θ s and θ o the dierences between the desired control inputs and the actual control inputs are used to design the TSM disturbance observers and the auxiliary systems, respectively. It is clear that these parameters will aect the disturbance estimate perormance and the auxiliary system states, which should be properly chosen to satisy the design requirement. 5 Numerical simulation To illustrate the eectiveness o the proposed TSM attitude control scheme o the NSV, the simulation results are presented in this section. The TSM attitude control schemes are designed in accordance with (28) (39). In this simulation, we suppose that the light velocity is V = 25 m/s and the light altitude is h = 3 km or the NSV. The initial values o the attitude angles and attitude angular velocities are assumed as α =2, β =1, μ =,andp = q = r = /s or the NSV. The saturation levels are chosen as v ri max = Nm and v li max = Nm, i =1, 2, 3. At the same time, the system uncertainties ΔF s (Ω) and ΔF (ω) are assumed as the 1% uncertainties in aerodynamic coeicients and

9 Chen M, et al. Sci China In Sci July 215 Vol : α c (º) α (º) β d (º) β (º) (a) μ d (º) μ (º) (c).8 (b) p (º/s) p d (º/s) (d) q (º/s) q d (º/s) (e) Figure 1 (e) pitch angular velocity, and () yaw angular velocity r (º/s) r d (º/s) () Control result o (a) angle o attack, (b) sideslip angle, (c) light-path roll angle, (d) roll angular velocity, aerodynamic moment coeicients, respectively. The external disturbances d s (t) andd (t) othensvare chosen as [7] and d s1 (t) =sin(.5t)+.25, d s2 (t) =2(sin(.2t)+.2)), d s3 (t) =1.5(sin(.5t)+.1)), (4) d 1 (t) = 15(sin(2t)+.1)) Nm, d 2 (t) = 2(sin(2t)+.2)) Nm, (41)

10 Chen M, etal. SciChinaInSci July 215 Vol : l ctrl (Nm).5.5 m ctrl (Nm) (a) (b) n ctrl (Nm) (c) Figure 2 (a) Roll, (b) pitching, and (c) yaw control moment. d 3 (t) = 2(sin(2t)+.2)) Nm. To design the TSM attitude controllers, all design parameters are chosen as q s = q s1 = q s2 = q = q 1 = q 2 = 9, p s = p s1 = p s2 = p = p 1 = p 2 = 5, K s = diag{2} 3 3, Λ s = diag{5} 3 3, Γ s = diag{.5} 3 3, M s = diag{15} 3 3, W s = diag{.5} 3 3, θ s = 5, B s = diag{5} 3 3, L s = diag{.5} 3 3, K = diag{2} 3 3, Λ = diag{3} 3 3, Γ = diag{.5} 3 3,M = diag{15} 3 3,W = diag{.5} 3 3, θ = 1, B = diag{1} 3 3,andL = diag{.5} 3 3. The desired attitude signals are taken as α d =2sin(.5t) 1 cos(.2t), β d =2,μ d =5,andp d = q d = r d =. Under the developed TSM attitude control schemes, the control results o the attitude angles and the attitude angular velocities are presented in Figure 1. From this igure, we observe that the satisactory tracking perormance is obtained or the desired attitude signals o the NSV with time-varying unknown disturbance and input saturation. At the same time, the corresponding control moments are shown in Figure 2, which are bounded and convergent. Based on the above simulation results, we conclude that the developed TSM attitude control scheme is valid or the NSV in the presence o unknown disturbance and input saturation. 6 Conclusion In this article, the TSM attitude control scheme has been studied or the NSV subject to unknown disturbance and input saturation. The TSMDO has been proposed to guarantee the inite time convergence o the disturbance estimate error. At the same time, an auxiliary design system has been designed to

11 Chen M, etal. SciChinaInSci July 215 Vol :11 compensate or the input saturation eect. Using the developed disturbance observer and the auxiliary system, the TSM attitude control scheme has been developed to improve the anti-disturbance ability and to speed up the convergence o all closed-loop system signals. Numerical simulation results have been presented to show the eectiveness o the developed TSM attitude control scheme under the integrated aection o the unknown compound disturbance and the input saturation. In uture, the developed TSM attitude control scheme based on the TSMDO can be extended to other nonlinear systems. Acknowledgements This work was supported by National Natural Science Foundation o China (Grant No ), Jiangsu Natural Science Foundation o China (Grant No. SBK21333), Program or New Century Excellent Talents in University o China (Grant No. NCET-11-83), Specialized Research Fund or the Doctoral Program o Higher Education (Grant No ), Aeronautical Science Foundation o China (Grant No ), and a Project Funded by the Priority Academic Program Development o Jiangsu Higher Education Institutions. Reerences 1 Xu B, Sun F C, Liu H P, et al. Adaptive kriging controller design or hypersonic light vehicle via back-stepping. IET Contr Theory Appl, 212, 6: Xu B, Shi Z K. An overview on light dynamics and control approaches or hypersonic vehicles. Sci China In Sci, 214, doi: 1.17/s Shi W, Jing Z L, Yang Y S. Ascent trajectory optimization or hypersonic vehicles via Gauss pseudospectral method. Int J Space Sci Eng, 213, 1: Xu B, Wang D W, Sun F C, et al. Direct neural discrete control o hypersonic light vehicle. Nonlinear Dyn, 212, 7: Gao D X, Sun Z Q. Fuzzy tracking control design or hypersonic vehicles via T-S model. Sci China In Sci, 211, 54: Xu B, Gao D X, Wang S X. Adaptive neural control based on HGO or hypersonic light vehicles. Sci China In Sci, 211, 54: Chen L, Jiang C S, Pu M. Online-SVR-compensated nonlinear generalized predictive control or hypersonic vehicles. Sci China In Sci, 211, 54: Chen M, Zhou Y L, Guo W. Robust tracking control or uncertain MIMO nonlinear systems with input saturation using RWNNDO. Neurocomputing, 214, 114: Xu B, Yang C G, Shi Z K. Reinorcement learning output eedback NN control using deterministic learning technique. IEEE Trans Neural Netw Learn Syst, 214, 25: Li Z J, Yang C G, Tang Y. Decentralised adaptive uzzy control o coordinated multiple mobile manipulators interacting with non-rigid environments. IET Contr Theory Appl, 213, 7: Xu B, Shi Z K, Yang C G, et al. Composite neural dynamic surace control o a class o uncertain nonlinear systems in strict-eedback orm. IEEE Trans Cybern, 214, 44: Yang C C, Zhai L F, Ge S S, et al. Adaptive model reerence control o a class o MIMO discrete-time systems with compensation o nonparametric uncertainty. In: Proceedings o American Control Conerence, Seattle, Xu B, Huang X Y, Wang D W, et al. Dynamic surace control o constrained hypersonic light models with parameter estimation and actuator compensation. Asian J Control, 214, 16: Du Y L, Wu Q X, Jiang C S, et al. Adaptive unctional link network control o near-space vehicles with dynamical uncertainties. J Syst Eng Electron, 21, 21: Chen M, Jiang B, Wu Q X, et al Robust control o near-space vehicles with input backlash-like hysteresis. Proc Inst Mech Eng I J Syst Contr Eng, 213, 227: Jiang B, Gao Z F, Shi P, et al. Adaptive ault-tolerant tracking control o near space vehicle using Takagi-Sugeno uzzy models. IEEE Trans Fuzzy Syst, 21, 18: Xu Y F, Jiang B, Tao G, et al. Fault tolerant control or a class o nonlinear systems with application to near space vehicle. Circuits Syst Signal Process, 211, 3: Xu B, Shi Z K, Yang C G. Composite uzzy control o a class o uncertain nonlinear systems with disturbance observer. Nonlinear Dyn, 215, 8: Yang J, Li S H, Yu X H. Sliding-mode control or systems with mismatched uncertainties via a disturbance observer. IEEE Trans Ind Electron, 213, 6: Yang J, Li S H, Sun C Y. Nonlinear-disturbance-observer-based robust light control or airbreathing hypersonic vehicles. IEEE Trans Aerosp Electron Syst, 213, 49:

12 Chen M, etal. SciChinaInSci July 215 Vol :12 21 Liu C, Chen W H, Andrews J. Tracking control o small-scale helicopters using explicit nonlinear MPC augmented with disturbance observers. Control Eng Practice, 212, 2: Chen W H, Ballance D J, Gawthrop P J, et al. A nonlinear disturbance observer or robotic manipulators. IEEE Trans Ind Electron, 2, 47: Chen M, Chen W H, Wu Q X. Adaptive uzzy tracking control or a class o uncertain MIMO nonlinear systems using disturbance observer. Sci China In Sci, 214, 57: Chen M, Ge S S. Direct adaptive neural control or a class o uncertain non-aine nonlinear systems based on disturbance observer. IEEE Trans Cybern, 213, 43: Li S H, Yang J. Robust autopilot design or bank-to-turn missiles using disturbance observers. IEEE Trans Aerosp Electron Syst, 213, 49: Guo L, Chen W H. Disturbance attenuation and rejection or systems with nonlinearity via DOBC approach. Int J Robust Nonlinear Contr, 25, 15: Du Y L, Wu Q X, Jiang C S, et al. Robust optimal predictive control or a near-space vehicle based on unctional link network disturbance observer. J Astronaut, 29, 3: Chen M, Jiang B. Robust attitude control o near space vehicles with time-varying disturbances. Int J Contr Automat Syst, 213, 11: Chen M, Wu Q X, Cui R X. Terminal sliding mode tracking control or a class o SISO uncertain nonlinear systems. ISA Trans, 213, 52: Zhou B, Gao H J, Lin Z L, et al. Stabilization o linear systems with distributed input delay and input saturation. Automatica, 212, 48: Lu P. Nonlinear systems with control and state constraints. Opt Contr Appl Methods, 1997, 18: Chen M, Ge S S, Ren B B. Adaptive tracking control o uncertain MIMO nonlinear systems with input saturation. Automatica, 211, 47: Chen M, Ge S S, How B. Robust adaptive neural network control or a class o uncertain MIMO nonlinear systems with input nonlinearities. IEEE Trans Neural Netw, 21, 21: Wang H Q, Chen B, Liu X P, et al. Robust adaptive uzzy tracking control or pure-eedback stochastic nonlinear systems with input constraints. IEEE Trans Cybern, 213, 43: Li Y M, Tong S C, Li T S. Adaptive uzzy output-eedback control or output constrained nonlinear systems in the presence o input saturation. Fuzzy Sets Syst, 214, 248: Wang H Q, Chen B, Liu X P, et al. Adaptive neural tracking control or stochastic nonlinear strict-eedback systems with unknown input saturation. In Sci, 214, 269: Lu P. Tracking control o nonlinear systems with bounded controls and control rates. Automatica, 1997, 22: Chen M, Wu Q X, Jiang C S, et al. Guaranteed transient perormance based control or near space vehicles with input saturation. Sci China In Sci, 214, 57: Wang J Y, Sun Z W. 6-DOF robust adaptive terminal sliding mode control or spacecrat ormation lying. Acta Astronaut, 212, 73: Zhang R M, Sun C Y, Zhang J M, et al. Second-order terminal sliding mode control or hypersonic vehicle in cruising light with sliding mode disturbance observer. J Control Theory Appl, 213, 11: Ni S B, Shan J Y. Smooth second-order nonsingular terminal sliding mode control or reusable launch vehicles. Int J Intell Comput Cybern, 214, 7: Man Z H, Yu X H. Terminal sliding mode control o MIMO linear systems. IEEE Trans Circuits Syst I-Fundamental Theory Appl, 1997, 44: Lin D, Wang X Y, Yao Y. Fuzzy neural adaptive tracking control o unknown chaotic systems with input saturation. Nonlinear Dyn, 212, 67: Hall C E, Shtessel Y B. Sliding mode disturbance observer-based control or a reusable launch vehicle. J Guid Control Dyn, 26, 29: