Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase

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Research article

Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase Yao Zhang, Shengjing Tang n, Jie Guo Key Laboratory of Dynamics and Control of Flight Vehicle, Ministry of Education, School of Aerospace Engineering, Beijing Institute of Technology, Beijing 100081, China

art ic l e i nf o

a b s t r a c t

Article history: Received 17 May 2016 Received in revised form 19 February 2017 Accepted 23 August 2017

In this paper, a novel adaptive-gain fast super-twisting (AGFST) sliding mode attitude control synthesis is carried out for a reusable launch vehicle subject to actuator faults and unknown disturbances. According to the fast nonsingular terminal sliding mode surface (FNTSMS) and adaptive-gain fast super-twisting algorithm, an adaptive fault tolerant control law for the attitude stabilization is derived to protect against the actuator faults and unknown uncertainties. Firstly, a second-order nonlinear control-oriented model for the RLV is established by feedback linearization method. And on the basis a fast nonsingular terminal sliding mode (FNTSM) manifold is designed, which provides fast finite-time global convergence and avoids singularity problem as well as chattering phenomenon. Based on the merits of the standard supertwisting (ST) algorithm and fast reaching law with adaption, a novel adaptive-gain fast super-twisting (AGFST) algorithm is proposed for the finite-time fault tolerant attitude control problem of the RLV without any knowledge of the bounds of uncertainties and actuator faults. The important feature of the AGFST algorithm includes non-overestimating the values of the control gains and faster convergence speed than the standard ST algorithm. A formal proof of the finite-time stability of the closed-loop system is derived using the Lyapunov function technique. An estimation of the convergence time and accurate expression of convergence region are also provided. Finally, simulations are presented to illustrate the effectiveness and superiority of the proposed control scheme. & 2017 Published by Elsevier Ltd. on behalf of ISA.

Keywords: Reusable launch vehicle Fast nonsingular terminal sliding mode Super-twisting algorithm Adaptive gain Fault tolerant

1. Introduction Reusable Launch Vehicles (RLVs) have gradually developed to be used repeatedly to access the space allowing for reducing the flight operation costs [1]. A multitude of favorable research has been investigated at NASA's Marshall Space Flight Center to improve safety, reliability and affordability for RLV [2,3]. Whereas the philosophy sounds interesting and attracting, a major challenge posed from outer space into earth's atmosphere is that of atmospheric reentry. During the reentry phase a plenty of stringent constraints and unknown uncertainties come into action [4]. Considering these inevitable issues, the control system requires to have capabilities of optimality, robustness and reconfiguration. The main goal of the reentry attitude controller is ensuring that the attitude angle tracks the guidance commands accurately in spite of the uncertainties and unknown external disturbances. Additionally, the safety of the RLV has been and will continue to be an important issue in the reentry phase [5]. Generally speaking, n

Corresponding author. E-mail addresses: [email protected] (Y. Zhang), [email protected] (S. Tang), [email protected] (J. Guo).

fault tolerant control has been a hot research area in the security control of the aircraft. The main design schemes for fault tolerant control are classified as either passive or active [6]. Passive schemes operate independently of any fault information and basically exploit the robustness of the underlying controller. Such schemes are usually less complex, but in order to cope with ‘worst case’ fault effects, are conservative. Active fault tolerant controllers react to the occurrence of faults, typically using information from a fault detection and isolation scheme, and invoke some form of reconfiguration. This represents a more flexible but complex architecture. Subsequent methods have tended to focus on online adaption or online controller synthesis [6]. Unsurprisingly, many robust control paradigms have been used as the basis for fault tolerant controllers. These include gain scheduling [7], adaptive control [8], H∞ [9], model-following [10], pseudo-inverse methods [11,12], nonlinear dynamic inversion (NDI) [13,14], multiple model approaches [15] and model prediction control (MPC) [16,17]. It is noted that these existing robust fault tolerant control methods can only provide asymptotical stability rather than finite-time convergence. It is widely known that besides faster convergence rate, sliding mode control (SMC) as a representative technology of the field of finite-time convergence theory, usually performs higher accuracies, better disturbance

http://dx.doi.org/10.1016/j.isatra.2017.08.012 0019-0578/& 2017 Published by Elsevier Ltd. on behalf of ISA.

Please cite this article as: Zhang Y, et al. Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.08.012i

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rejection features and strong fault tolerance The possibilities of exploiting the inherent robustness properties of sliding mode for fault tolerance have previously been explored for aerospace applications and some works in [18–24] have argued that SMC has the potential to become an alternative to reconfigurable control. To develop these outstanding characteristics of SMC such as faster convergence, robustness to unknown uncertainties and strong capability of resistance to nonlinear fault tolerant systems, some efforts, such as terminal SM control (TSMC) method [25,26], nonsingular TSMC (NTSMC) approach [27], fast TSMC (FTSMC) technique [28] and nonsingular FTSMC (NFTSMC) law [29]. Additionally, another research issue not to be tackled in current SMC algorithms is the undesirable ‘chattering’ phenomenon due to the discontinuous control action. The most traditional way for reducing chattering is to replace the discontinuous control function by “saturation” or “sigmoid ones” [29]. Although this approach yields continuous control and chattering elimination, it constrains the system's trajectories not to the sliding mode (SM) surface but to its vicinity losing the robustness to the uncertainties. Recently, a new approach called higher order SMC (HOSMC) has been proposed to attenuate the chattering effect [30–33]. Using the HOSM control method allows driving to zero the sliding variable and its consecutive derivatives in the presence of the unknown disturbances/uncertainties and thereby, can improve on the accuracy of the sliding variable stabilization [34]. Several favorable attempts to apply this technique to hypersonic vehicle flight control [35,36] have been undertaken. However, one of the disadvantage of imposing an r-th order SMC on the vehicle is the necessity of having s, s,̇ s¨,…, s(r-1) (where s is the sliding variable) available. Second order sliding mode control (SOSMC), as one special case of HOSMC, does not require the derivative information. At present, the super-twisting (ST) control algorithm is one of the most powerful SOSMC algorithms that can handle a relative degree equal to one. Generally, it generates the continuous control function that steers the sliding variable and its derivative to zero in finite time in the presence of the smooth uncertainties with bounded gradient, when this boundary is known. Since ST algorithm includes a discontinuous control function under the integral, chattering is not eliminated but attenuated. Moreover, ST algorithm can obtain superior rapidity and stability to HOSM controller and prevent from acquiring the high-order derivative [37]. However, Successful implementation of ST sliding mode controller for the fault tolerant control problem of the RLV requires the knowledge of the boundaries of the total fault and disturbance information [38,39]. In many practical cases, this boundary cannot be easily estimated. In order to maintain the robustness of the closed-loop system, the overestimating of the boundaries yields to larger control gains and leads to a higher chattering amplitude. Hence, a comprehensive study of the ST control algorithm with adaptation has been undertaken. The adaptive-gain fast ST (AGFST) control law, which handles the perturbed system with the additive uncertainty/disturbance of certain class with the unknown boundary, was proposed in [40]. More recently, two approaches to adapt the gain have been developed for the ST algorithm. The first attack is to reconstruct the uncertainty/disturbance and to adapt the gain according to its estimated value, as discussed in [41]. Although this method can track the required values of gains very accurately, it requires the knowledge of the bound of the derivative of the uncertainty. The second approach to adapt the controller gain is to detect the sliding mode: increasing the gain until the sliding surface is reached and decreasing the gain when the sliding mode has been established [42,43].

The motivation of the research is to develop a robust fast terminal sliding mode control (RFTSMC) method with AGFST algorithm for the fault tolerant control problem of the RLV. Compared with existing control methods, the main contributions of this paper are summarized as follows: i) the presented RFTSMC based on AGFST algorithm resolves the main obstacles for application of SMC, that is, chattering and high activity of control action. Furthermore, the adaptation with respect to the control gains overcomes the overestimating drawback and resolves robust gains being determined difficultly. ii) The proposed AGFST algorithm contains the merits of standard ST algorithm, constant and plus power rate reaching laws and adaption. It does not require any prior knowledge of model uncertainties, external disturbances and actuator faults; and provides strong robustness against the uncertainty/disturbance growing in time or together with the RLV tracking error. Additionally, it has faster convergence rate than standard ST algorithm. iii) The presented RFTSMC based on AGFST algorithm is non-singularity and improves the convergence rate when the tracking error is far away from the origin. It will yield to be applied in practical systems. iv) The finite-time rigorous convergence is demonstrated, and the estimation of the convergence time and accurate expression of convergence region are also put forward. The paper is organized as follows. In Section 2, the vehicle model with fault tolerant problem is formulated. The second-order nonlinear control-oriented model is established by feedback linearization in Section 3. In Section 4, the RFTSMC system is developed based on AGFST control algorithm, and the finite time stability of the closed-loop system with the proposed control law is analyzed. Illustrative simulations of the proposed control approach for the RLV and concluding remarks are given in Section 5 and Section 6, respectively.

2. Model description The equations for six-degree-of-freedom rigid reentry flight vehicle are described in [35]. The motion of this vehicle can be divided into translational motion and rotational motion. The dynamic equations of rotational motion of a reentry vehicle are given by [35,36]

⎡ ṗ ⎤ ⎧ ⎡ p⎤ ⎛ ⎡ p⎤⎞ ⎫ ⎢ ⎥ −1⎪ ⎢ ⎥ ⎜ I¯⎢ q ⎥⎟ + M + ΔM ⎪ ¯ ⎨ ⎬ I = − × q ̇ c ⎢ q⎥ ⎪ ⎢⎣ r ⎥⎦ ⎜⎝ ⎢⎣ r ⎥⎦⎟⎠ ⎪ ⎢⎣ r ̇ ⎥⎦ ⎩ ⎭

(1)

where p, q and r represent the roll, pitch and yaw angular rates, respectively. Mc =[Ml,Mm,Mn]T is the control torque vector, in which Ml , Mm and Mn are the roll, pitch and yaw control torques defined in the body frame, respectively. ΔM denotes the unknown bounded external disturbance moment. I ̅ =I +∆I stands the inertial tensor in the body frame with ΔI ∈ R 3 × 3 as the uncertain part of the inertia matrix. The nominal inertial tensor I is defined as

⎡I 0 −Izx ⎤ ⎢ x ⎥ Iy 0 ⎥ I = ⎢0 ⎢ ⎥ ⎣ −Izx 0 Iz ⎦ due to mass symmetry about the x-axis in the body frame. The kinematic equations of reentry vehicle are defined as [35,36]:

Please cite this article as: Zhang Y, et al. Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.08.012i

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T ϑ = ⎡⎣ ϑl( t ), ϑm( t ), ϑn( t )⎤⎦ and satisfies ϑ ≤ lϑ , lϑ is an unknown positive constant, where ϑv is defined as follows

α̇ = − p cos α tan β + q − r sin α tan β sin σ ⎡ + ⎣ ψ ̇ cos γ − ϕ ̇ sin ψ sin γ cos β

⎧ sgn(M )M max − M (t ), v v v ϑv(t ) = ⎨ ⎩ 0,

+ ( θ ̇ + ωe )( cos ϕ cos ψ sin γ − sin ϕ cos γ )⎤⎦ cos σ ⎡ ⎤ − ⎣ γ ̇ − ϕ ̇ cos ψ − ( θ ̇ + ωe )cos ϕ sin ψ ⎦ cos β β ̇ = p sin α − r cos α + sin σ ⎡⎣ γ ̇ − ϕ ̇ cos ψ − ( θ ̇ + ωe )cos ϕ sin ψ ⎤⎦ + cos σ ⎡⎣ ψ ̇ cos γ − ϕ ̇ sin ψ sin γ + ( θ ̇ + ωe )( cos ϕ cos ψ sin γ − sin ϕ cos γ )⎤⎦ σ ̇ = − p cos α cos β − q sin β − r sin α cos β + α̇ sin β − ψ ̇ sin γ − ϕ ̇ sin ψ cos γ + ( θ ̇ + ωe )( cos ϕ cos ψ cos γ − sin ϕ sin γ )

(2)

where α , β , and σ are AOA, sideslip angle and bank angle, respectively. γ denotes flight path angle and ψ denotes heading angle. θ and ϕ are longitude and latitude of the reentry vehicle. ωe is the angular rate of Earth-rotation. As known, the attitude controller design is the key of the research. When the guidance law is determined, the rotational motions of the RLV are much faster than translational motions, the time derivatives of both the position and the velocity are considered to be negligible with respect to the rotational motion. Therefore, the rotational equations of motion (1) and (2) can be further simplified as follows [35,36]

⎧ ̇ ⎪ Ω = Rω + Δf ⎨ ⎪ −1 × −1 ⎩ ω̇ = − I ω Iω + I Mc + Δd where,

Ω = ⎡⎣ α , β , σ ⎤⎦

T

is

(3) the

aerodynamic

angle

vector,

T ω = ⎡⎣ p, q, r ⎤⎦ is the attitude angular rate vector, R ∈ R 3 × 3 is the T

coordinate-transformation matrix, Δf = ⎡⎣ Δf1 , Δf2 , Δf3 ,⎤⎦ denotes the unknown bounded uncertainties caused by the model reduction, ω× ∈ R 3 × 3 stands for the skew-symmetric matrix operator on vector ω, and Δd ∈ R 3 denotes the bounded uncertain term. R , ω×, and Δd are given by

⎡ − cos α tan β 1 − sin α tan β ⎤ ⎢ ⎥ R=⎢ sin α 0 − cos α ⎥ ⎢⎣ − cos α cos β − sin β − sin α cos β ⎥⎦

3

(4)

⎡ 0 −r q ⎤ ⎢ ⎥ ω× = ⎢ r 0 −p⎥ ⎢⎣ q p 0 ⎥⎦

(5)

Δd = I −1⎡⎣ −ΔIω̇ − ω× ΔIω + ΔM ⎤⎦

(6)

Mv ≥ Mvmax otherwise

Two types of actuator faults existing in practical system [44,45] T are taken into account: Fδ=⎡⎣ Fδ1,Fδ2,Fδ 3⎤⎦ is the additive fault, which means that faults enter the system in an additive way satisfying Fδ ≤ lδ , lδ is an unknown positive constant. D=diag(δl,δm,δn) is the actuator effectiveness which satisfies 0 ≤δv≤1. Note that the case δv = 1 means that the v-channel actuator works normally; if δv = 0, the v-channel actuator has failed completely without any control torque supplied; and 0 <δv<1 corresponds the case in which the vchannel actuator has partially lost its effectiveness, but it still works all of the time. Note that the composed forms of actuator faults considered in this paper are actuator saturation, additive faults and partial loss of actuator effectiveness. Therefore, the reentry attitude dynamics of the vehicle with actuator faults and actuator constraints can be expressed as

⎧ Ω̇ = Rω + Δf ⎨ −1 × −1 ⎪ ⎩ ω̇ = − I ω Iω + I ⎡⎣ D sat(Mc ) + Fδ⎤⎦ + Δd ⎪

ω̇ = − I −1ω× Iω + I −1⎡⎣ D sat(Mc ) + Fδ⎤⎦ + Δd

(7)

(9)

The aim of this paper is to determine the control torque Mc so T that the aerodynamic angles Ω = ⎡⎣ α , β , σ ⎤⎦ can follow the guiT

dance commands Ωc = ⎡⎣ αc , βc , σc ⎤⎦ in the presence of inertia uncertainties, external disturbances, actuator faults and saturations. Specifying the control torque vector Mc in equation set (3), which leads the output vector Ω to track the attitude command Ωc in a finite time.

lim‖e‖ = lim‖Ω − Ωc ‖ = 0 t > tF

t > tF

(10)

where e = Ω − Ωc is the tracking error.

3. Feedback linearization By the selection of control input as control torque vector Mc , and the output as attitude angle vector Ω , The vector relative degree of system (9) is ( 2, 2, 2). After differentiating output vector Ω twice, the control input vector Mc appears.

d Ω¨ = ( Rω + Δf ) = = F + GMc + ΔD dt

In order to design an active fault tolerant control strategy for the reentry vehicle, we will consider the case of existing inertia uncertainties, external disturbances, actuator faults and saturations. Consider the reentry attitude system with actuator faults and actuator constraints

(8)

(11)

where F, G, and ΔD are introduced to denote the normal part, the coupled part, and the lumped uncertainty and disturbance, respectively. The expressions of F, G, and ΔD are

(

)

F = Ṙ − RI −1ω× I ω + Δf ̇

(12)

G = RI −1

(13)

ΔD = RI −1⎡⎣ D ϑ + Fδ + (D − I )Mc ⎤⎦ + RΔd

(14)

where sat (Mc )=[sat(Ml ),sat(Mm),sat(Mn)]T is the vector of actual control torques generated by the actuators (or thrusters), in which sat( Mv ), v ¼ l, m, n denotes the nonlinear saturation characteristic of the actuators and is of the form sat( Mv )=sign( Mv )min Mvmax, Mv . It is also expressed as

According to Eq. (10), the following double integral tracking error dynamic system

sat( Mv )=ϑv+Mv . The part of excess saturation limited is given by

̇ Ω̇ -Ωċ . is established, where x1=e =Ω − Ωc and x 2=e =

{

}

⎧ x1̇ = x 2 ⎨ ⎩ ẋ2 = F + GMc − Ω¨ c + ΔD ⎪

⎪

(15)

Please cite this article as: Zhang Y, et al. Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.08.012i

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uncertainty/disturbance.

4. RFTSMC system design based on AGFST algorithm In this section, the RFTSMC technique based on AGFST algorithm (AGFST-RFTSMC) is proposed for the reentry vehicle. The structure of the proposed attitude control scheme is presented in Fig. 1. 4.1. Improved fast nonsingular terminal sliding mode manifold To avoid the singularity problem and improve the convergence rate when the error is far away from the origin, an IFNTSM manifold is developed for the fault tolerant attitude control problem of the RLV on the basis of the technique merits of the FTSMC method and NTSM method. Introduce the following modified nonlinear function ⎧ p/q ⎪X , ϕ (X ) = ⎨ p p ⎪ (2 − p/ q )ε q − 1X + (p/ q − 1)ε q − 2sgn(X )X 2 , ⎩

if S¯ = 0 or S¯ ≠ 0, if S¯ ≠ 0, X < ε

The following theorem with regard to the IFNTSM manifold defined by (17) can be obtained. Theorem 1. Consider the uncertain system (15) and IFNTSM manifold S defined by (17), if the sliding manifold S =S ̅ =03×1 is reached for all t ≥ts≥0 , then x1i can converge to zero in finite time. Proof. If the sliding manifold S = S ̅ =03×1 is reached for all t ≥ts≥0, the sliding manifold can be rewritten in scalar forms:

x2i + λ1ix1i + λ2ix1i ri = 0 hence from (20)

x2i = − λ1ix1i − λ2ix1i ri

3

W=

∑ Wi = i=1

where S¯ = Ẋ + λX p / q , ε is a small positive constant, sgn( ∙ ) denotes the signum function. The positive odd integers p and q are chosen such that 0 o p/q o 1. The IFNTSM manifold S ∈R 3 for the reentry vehicle is defined as

Si = x2i + λ1ix1i + λ2iSaui,

i = 1, 2, 3

(17)

where λ1i and λ2i are positive constants, and the auxiliary sliding manifold Saui is defined as r ⎧ ⎪ x1i i, Saui = ⎨ ⎪ r −2 2 r −1 ⎩ (2 − ri )ε i i x1i + (ri − 1)ε i i sgn(x1i )x1i ,

if S¯ i = 0 or S¯ i ≠ 0, if S¯ i ≠ 0, x1i < ε i

x1i ≥ ε i

(18)

with S¯i = x2i + λ1ix1i + λ2ix1i ri and ri=pi /qi . The first time derivative of the proposed IFNTSM manifold Si is described as Eq. (19), which is a continuous function. Note that there exists no singularity problem in the preceding equation.

⎧ ri − 1 ⎪ ẋ x2i , i 1i 2i + λ1ix1i + λ2irx Si̇ = ⎨ ⎪ r −1 r −2 ⎩ x2̇ i + λ1ix1i + λ2i[(2 − ri )εi i x1i + 2(ri − 1)εi i sgn(x1i )x1i x2i ],

Remark 1. In this paper, we pay our attention into solving the singularity problem in the FTSM controller and conquering the undesirable phenomenon related to the convergence rate when using the NTSM manifold. Compared with the classical NTSM manifold, the presented IFNTSM manifold includes two necessary terms: the term λ1ix1i can play an important role in accelerating the convergence speed when the state keeps far away from the equilibrium point, while the term λ2iSaui can guarantee fast convergence of the system state when appearing near the

⎧ ri − 1 ⎪ ΔD + λ x x2i , i i 1i 1i 1i + λ2irx Di = ⎨ ⎪ r −1 r −2 ⎩ ΔDi + λ1ix1i + λ2i[(2 − ri )εi i x1i + 2(ri − 1)εi i sgn(x1i )x1i x2i ],

sliding mode manifold. Therefore, the proposed IFNTSM manifold contains the prominent advantage of the FTSM and NTSM manifold, and it can improve the global convergence speed and enhance the system robustness against the unknown

(21)

For the system (15), consider as a Lyapunov function candidate

X ≥ε

(16)

(20)

1 2

3

∑ x12i

(22)

i=1

Differentiating the expression in (22), we obtain

Ẇi = − λ1ix12i − λ2ix1i ri+ 1 = − 2λ1iWi − 2

ri + 1 ri + 1 2 λ2iWi 2

≤0

(23)

Therefore from (23)

Ẇi + 2λ1iWi + 2

ri + 1 ri + 1 2 λ2iWi 2

≤0

(24)

Because 0
λ V 1 ts ≤ ln 1i λ1i(ri + 3)

if S¯i = 0 or S¯i ≠ 0, if S¯i ≠ 0,

1 − ri 2 (t0)

2

+2

ri − 1 2 λ2i

ri − 1 2 λ2i

(25)

x1i ≥ εi

x1i < εi

(19)

Finally x1i can converge to zero in finite time as claimed. 4.2. Attitude fault tolerant controller design based on AGFST algorithm Substituting (15) into (19), we have the following sliding manifold dynamic equation:

S ̇ = F + GMc − Ω¨ c + D

(26)

where D ¼ [D1,D2,D3]T and Di (i ¼ 1,2,3) is given by

if S¯i = 0 or S¯i ≠ 0, if S¯i ≠ 0,

x1i ≥ εi

x1i < εi

(27)

with S¯i = x2i + λ1ix1i + λ2ix1i ri . Note that the sideslip angle β ≈ 0 during reentry, we have

Please cite this article as: Zhang Y, et al. Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.08.012i

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4.3. Stability analysis To proceed to analyze the Lyapunov stability for the closedloop system (34), Firstly, we are required to demonstrate the boundedness of the adaptive gains ρ(t ) and ρ¯ (t ). Fig. 1. The structure of the proposed AGFST-RFTSMC system for the reentry vehicle.

det(G ) =

cos β − sin β tan β 1 ≈ ≠0 2 2 IxIz − Izx IxIz − Izx

(28)

Proposition 1. Considering the uncertain system (30), the AGFST algorithm is designed according to (31) with the adaptive gains (32) and (33). Then, the solution of the closed-loop system (34) is uniformly bounded, i.e. S , U , ρ(t ) and ρ¯ (t ) are uniformly bounded. Proof. Consider as a Lyapunov function candidate

Therefore, the system (26) can be linearized completely without zero dynamics by using the following feedback control law

Mc = G −1 −F + U + Ω¨ c

(

)

(29)

V1 =

Ṡ = U + D

(30)

3

V1̇ =

∑ Si(Ui + Di)

then substituting the AGFST algorithm (31) with χ(t ) ≡ 0 into (36) yields 3

Assumption 1. Assume that the total unknown uncertainty Di in (30) has the bounded first-order derivative and satisfies Di̇ ≤ c1 + c2‖x 2‖ + c3‖x 2‖2 or unknown scalar bounds ci>0 (i=1,2,3). The aim of the controller design is to drive the proposed sliding mode manifold Si and its derivative Si̇ to zero in finite time in the presence of the unknown uncertainty with the unknown boundary by means of continuous control without the control gain overestimation. The expression of the proposed AGFST algorithm can be obtained by using

Ui(t ) = − ρi (t )ψ0i(t ) −

χi (t ) =

1/2

∫0

t

ρ¯i (τ )χi (τ )dτ

sgn(Si ) + Si

1 3 1/2 Si sgn(Si ) + Si sgn(Si ) + 2 2

(31)

with the adaptive gains are selected as μ1 ⎧ + sni (Si ) μ2), ⎪ a 0i(sni (Si ) ρi̇ (t ) = ⎨ ⎪ ⎩ γi Si ,

ρi (t ) ≥ νi ρi (t ) < νi

(32)

ρ¯i (t ) = 2δiρi (t )

(33)

μj

μj

where sni(Si ) =sgn( Si − εi) Si (j=1,2), and a0i,γi,δi are positive design parameters. Besides, μ1≥1, 0<μ2<1,ρi ( 0)≥ρm , and νi,εi,ρm are a range of arbitrary small positive constants. Substituting the AGFST control algorithm (31) into (30) yields a particular case of the system

⎧ S ̇ = − ρ(t )ψ (t ) + ξ ⎨ ⎪ ⎩ ξ ̇ = − ρ¯ (t )φψ (t ) + Ḋ ⎪

(34)

where ρ(t ) = diag(ρi (t )), ψ(t ) = sgn1/2(S ) + S , sgn1/2(S ) = [ Si

ξ = D − χ (t ) , 1

Ḋ = [Di̇ ]T ,

1/2

sgn(Si )]T ,

T

⎡ t ⎤ χ (t ) = ⎢⎣ ∫ ρ¯i (τ )χi (τ )dτ ⎦⎥ , 0

(36)

i=1

V1̇ ≤ −

ψ0i(t ) = Si

(35)

Differentiating the expression in (35) yields

T

where U ¼ [U1,U2,U3] is selected as the new manipulated control input in this paper. Substituting (29) into (26), the basic model for attitude controller design can be obtained by

1 T S S 2

∑ ρi i=1

(S

i

3/2

+ Si

2

3

)+∑ S M i

i=1

D

(37)

where MD ≥ Di . Therefore, it is easy to verify V1̇ ≤ 0 if the inequality Si ≥εs are satisfied where

⎛ ⎧ ⎪ M D εs = max⎜⎜ εi , min⎨ , ⎪ ⎩ ρi ⎝

⎞ ⎪ MD ⎫ ⎟ ⎬ ⎪⎟ ρi ⎭⎠

(38)

When Si >εi , the characteristic can be obviously obtained from (32) that the control gain ρi can be dynamically increased until the sliding manifold Si converges to the prescribed small interval εi around the origin. Then the gain shall start reducing. If the inequality ρi ≤νi is satisfied, the adaptive system behavior is given by the second equation in (32). Note that ρi ≤νi is established when Si=0. Otherwise, this gain reduction shall be reversed as soon as the sliding manifold Si starts deviating from the domain Si ≤εi . Therefore, the adaptive gain ρi is a positive function coupled with a small fluctuation transition period around the arbitrary small positive constant νi . From the proposed control law (31), the inequality -χi <0 (or -χi >0) is true as long as Si>0 (or Si<0). Define χ(t ) ≠ 0, due to introducing the integral feedback that contains -χi (t ) into (31), the proposed control law provides more precise control performance and ensures the sliding manifold Si to reach an arbitrarily smaller neighborhood εm (i.e. εm<εs ) near the origin. Accordingly, the proposed controller is a bounded function. Si,Uiandρi are thereby uniformly bounded. Besides, the adaptive gain ρ¯i is also uniformly bounded from the aforementioned analysis. Proposition 1 is proven. Hence the main result of this paper is formulated in the following theorem. Theorem 2. Consider the closed-loop nonlinear system (34) and the proposed IFNTSM manifold (17). Suppose that the unknown uncertainty Di satisfies Assumption 1 for some unknown gains ci>0 (i=1,2,3). Then, for any initial conditions S0 and ξ0 , there exists a finite time 0 4 tF and a positive parameter εi (as soon as the condition

ρ¯ (t ) = diag(ρ¯i (t )),

−1

φ = diag( 2 Si 2 ) + I3. Note that the differential equations in (34) have discontinuous right hand sides. The solutions to such equations must therefore be understood in the sense of Filippov [46].

ρi >

(2ς1 + 4zi + κi2)2 − 4κi2ς1 2κi(8zi2 + κi2)

(39)

holds. If Si(0) >εi ) so that the proposed IFNTSM manifold, i.e.

Please cite this article as: Zhang Y, et al. Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.08.012i

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Si ≤Sε , is established ∀t ≥tF and the errors x1 and x 2 can separately converge to small regions E1 and E2 of the origin via the proposed AGFST-RFTSMC method (31) with the adaptive gains (32) and (33), where

⎧ S = max(ε , ε , ε ) ρi ρ¯ i i ⎪ ε ⎪ ⎛ ⎪ S S ⎨ E1: = max⎜⎜ ε, ε , ri ε 2λ2i ⎪ ⎝ 2λ1i ⎪ r ⎪ ⎩E : = S + λ E + λ E i 2

ε

1i 1

⎡ diag(4λ + κ 2) diag( − κ )⎤ ⎡ P1 P2 ⎤ i i i ⎥ ⎥ P=⎢ :=⎢ ⎥⎦ ⎢⎣ diag( − κi ) ⎣ P2 P3⎦ 2I3

⎞ ⎟⎟ ⎠ (40)

with elements

(41)

Proof. A proof of Theorem 2 is presented as bellows and the proof is split into two steps. Step 1 In the first step, we will present system (34) in a form convenient for the Lyapunov analysis. In order to do this a new state vector is constructed T ζ = ⎡⎣ ψ (t ), ξ⎤⎦

(42)

and the system (34) can be rewritten as

{ (S, ξ) ∈ R : S = 0} 6

(51)

̇ V̇ = Vζ̇ + Vad

(52)

From (47), (48) and (50), the first term of (52) is written as T T T Vζ̇ = ζ T (t )Pζ (̇ t ) + ζ ̇ (t )Pζ (t ) = ζ T (t )PA¯ (t )ζ¯(t ) + ζ¯ (t )A¯ (t )Pζ (t )

= − ζ T (t )Q¯ (t )ζ¯(t ).

(53)

The symmetric matrix Q̅ is defined

⎡ Q¯ Q¯ 21 ⎤ 11 ⎥ Q¯ = ⎢ ⎢⎣ Q¯ Q¯ 22 ⎥⎦ 12 with

⎡ φ( − ρ(t )ψ (t ) + ξ )⎤ ⎥ ζ̇ = ⎢ ⎣ −ρ¯ (t )φψ (t ) + Ḋ ⎦

(54)

Q¯ 11: = 2P1ρ(t ) − 2P2( − ρ¯ (t ) + ς (t )) − κiI3,

Q¯ 21 = Q¯ 12: = P2 ρ(t )

−P3( − ρ¯ (t ) + ς (t )) − PI 1 3 , Q̅ 22 = − 2P2I3 − κiI3. In order to guarantee

(43)

Due to Assumption 1, the time derivative of the uncertainty Ḋ is bounded. Then suppose that there exists a positive constant ς1 2

such that ς1>2lF ( lF ≔c1 + c2 x 2 +c3 x 2 ), we can obtain 1 1 1 3 Di̇ ≤ c1 + c2‖x 2‖ + c3‖x 2‖2 ≤ ς1 ≤ ς1 + ς1 Si 2 + ς1 Si 2 2 2

(44)

{ ( S )∈R |S =0}, then from the third

Define the subspace S0=

3

equation of (31) there exists

1 3 + Si 2 2

Ξ=

then the function V( ∙ ) in (49) is everywhere continuous, and differentiable everywhere except on the subspace Ξ . Furthermore, it is worth noting that the Lyapunov function V is positive definite and radially unbounded. The derivative of the Lyapunov function is presented

⎧ ⎛ ¯ ¯⌢ ¯ ¯ ⌢⎞ ⎪ ε = min⎜ μ1 θ2i + θ1i ρi , μ2 θ2i + θ1i ρi ⎟ ⎜ ⎟ ⎪ ρi a 0i a 0i ⎝ ⎠ ⎪ ⎨ ⌢ ⌢ ⎛ ⎪ θ¯ + θ¯1i ρ¯i μ θ¯2i + θ¯1i ρ¯i ⎞⎟ ⎪ ε ρ¯ i = min⎜ μ1 2i , 2 ⎜ 2δia 0i 2δia 0i ⎟⎠ ⎪ ⎝ ⎩

χi (t ) =

(50)

where λ i > 0, κi = 2δi > 0. Define the subspace

2i 1

1 2

3 ⌢ 1 where Vζ = ζ T (t )Pζ (t ), Vad= 2 ∑i=1 (ρĩ 2 +ρi̅ ̃ 2 ), ρ˜i = ρi − ⌢ ρi , ρ¯˜i = ρ¯i − ρ¯i , ⌢ and ⌢ ρi > 0, ρ¯i > 0 are some constants. The positive definite matrix P is constructed as

+ Si

(45)

3

for arbitrary S ∈R / S0 . Hence, from (44)

Di̇ ≤ ς1 χi (t )

(46)

the positive definiteness of the matrix Q̅ , using Schur's lemma, the matrix Q̅ is required to satisfy the following conditions:

⎧ Q¯ > 0 ⎪ 11 ⎪ ¯ ⎨ Q 22 > 0 ⎪ ¯ ¯ ¯ −1 ¯ T ⎪ ⎩ Q 11 − Q 12Q 22 Q 12 > 0

(55)

Thus, the matrix Q̅ will be positive definite with a minimal

( )

eigenvalue λ min Q̅ ≥κi if

ρi >

(2ς1 + 4zi + κi2)2 − 4κi2ς1 2κi(8zi2 + κi2)

(56)

In view of (53) and supposing that (56) holds, it is easy to show that 3

Accordingly, there exists the function

ς0i(t ) ≤ς1 such that

Di̇ =ς0i(t )χi (t ). Due to χi (t )=φiψ0i(t ), the system (43) can be expressed as

⎡ φ( − ρ(t )ψ (t ) + ξ ) ⎤ ζ̇ = ⎢ ⎥ = A¯ (t )ζ¯ ⎣ −ρ¯ (t )φψ (t ) + ς (t )φψ (t )⎦

(48)

Step 2 In the second step of the proof, the stability analysis of system (47) is performed. In order to do it the following Lyapunov function candidate is introduced

V = Vζ + Vad

(49)

(57)

From Lemma 4 in Ref. [47], we have 1

(47)

where ς (t ) = diag(ς0i(t )), ζ¯ = [φ, φ]ζ , and the matrix A¯ (t ) is defined by

⎡ I3⎤ −ρ(t ) ⎥ A¯ (t ) = ⎢ ⎣ −ρ¯ (t ) + ς (t ) 0 ⎦

1 −1 Vζ̇ ≤ − 2δminζ T (t )ζ¯(t ) = − 2δmin ∑ σi 2 (ψ02i + ξi2) − 2δmin‖ζ‖2 2 i=1

Vζ̇ ≤ − δmin‖S‖− 2 ‖ζ‖2 − 2δmin‖ζ‖2

(58)

where δmin=min{ δi}.

3

‖ψ‖2 = ‖S‖2 +

∑ ( Si

3

+ 2 Si 2 )

i=1

(59)

and

λ min(P )‖ζ‖2 ≤ ζ T (t )Pζ (t ) ≤ λ max(P )‖ζ‖2

(60)

there exists

Please cite this article as: Zhang Y, et al. Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.08.012i

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‖S‖1/2 ≤ ‖ψ‖ ≤ ‖ζ‖ ≤

Vζ1/2 1/2 λ min (P )

(61)

1 −‖S‖− 2

≤−

1/2 λ min (P )

Vζ1/2

(62)

Then from (58) and (62), we have

Vζ̇ ≤ − η1Vζ − η2Vζ1/2

(63)

1/2 where η1 = 2δmin/λmax (P ), η2 = δminλ min (P ) /λ max(P ). ⌢ In view of Proposition 1, there exist positive constants ⌢ ρi and ρ¯i ⌢ ⌢ such that ρi − ρi < 0 and ρ¯i − ρ¯ii < 0. Then substitute Eqs. (32), (33) and (63) into Eq. (52), the equation can be reduced to the following

3

V̇ ≤ − η1Vζ − η2Vζ1/2 −

∑(

ρ˜i ρi̇ + ρ¯˜i ρ¯i̇

)

(64)

i=1

⌢ Due to the definition of ρ˜i and ρ¯˜i , we have ρ˜i ≤ ⌢ ρi and ρ¯˜i ≤ ρ¯i . Next, we can obtain 3

V̇ ≤ − η1Vζ − η2Vζ1/2 −

3

∑ θ¯1i(ρ˜i2 + ρ¯˜i2 ) i=1

−

∑ θ¯2i(

ρ˜i + ρ¯˜i

i=1

)

∑(

ρ˜i ρi̇

i=1

3

)

(

)

i=1

3

∑ θ¯2i(

3

ρ˜i + ρ¯˜i

) − ∑ ( ρ¯˜ ρ¯ ̇ )

i=1

i

i=1

i

(65)

3

∑ θ¯1i(ρ˜i2 + ρ¯˜i2 )

3

ρ˜i + ρ¯˜i

i=1 3

−

∑

)

ρ˜i a 0i(sni (Si ) μ1 + sni (Si ) μ2) − θ¯1i⌢ ρi − θ¯2i

(

)

3

V̇ ≤ − η1Vζ − η2Vζ1/2 −

∑(

)

ρ˜ii γi Si + ρ¯˜i 2δiγi Si ≤ 0

i=1

(70)

Therefore, there exists the condition Si(t ) ≤ εi such that the above Eq. (70) is established and the states x1 and x 2 are uniformly bounded. Then the proposed IFNTSM manifold, i.e. Si ≤Sε , is established ∀t ≥tF via the proposed AGFST algorithm (31) with the adaptive gains (32) and (33). Now, we have to start to discuss the characteristic of the convergence for x1 and x 2 . The analysis process is similarly divided into two cases.

3

ρ¯i − θ¯2i − ∑ ρ¯˜i 2δia 0i(sni (Si ) μ1 + sni (Si ) μ2) − θ¯1i⌢

(

i=1

)

(66)

(71)

Then (71) can be rewritten

⎛ ⎛ φ ⎞ φ ⎞ x2i + ⎜ λ1i − κi ⎟x1i + ⎜ λ2i − κi r ⎟x1i ri = 0 2x1i ⎠ 2x1i i ⎠ ⎝ ⎝

(72)

According to Theorem 1, the error x1i shall converge to a small interval E1 near the origin in finite time if the following inequalities

φκi 2x1i φκi 2x i ri

>0 >0

(73)

⎛ S E1: = max⎜⎜ ε, ε , λ1i 2 ⎝

ri

Sε ⎞ ⎟⎟ 2λ2i ⎠

(74)

Case 2 If x1i ≤ε , the error x1i shall converge to the small region E1 of zero. From (64), we obtain x2i ≤ E2 where the convergence region E2 is defined by

Select Sε = max(ερi , ε ρ¯ i , εi ), where

⎧ ⎛ ¯ ¯⌢ ¯ ¯ ⌢⎞ ⎪ ε = min⎜ μ1 θ2i + θ1i ρi , μ2 θ2i + θ1i ρi ⎟ ⎜ ⎟ ⎪ ρi a a 0i 0i ⎝ ⎠ ⎪ ⎨ ⌢ ⌢ ⎛ ⎪ θ¯ + θ¯1i ρ¯i μ θ¯2i + θ¯1i ρ¯i ⎞⎟ ⎪ ε ρ¯ i = min⎜ μ1 2i , 2 ⎜ 2δia 0i 2δia 0i ⎟⎠ ⎪ ⎝ ⎩

φκi ≤ Sε

are satisfied, where the convergence region E1 is given by

i=1

E2: = Sε + λ1iE1 + λ2iE1ri

(75)

(67)

As a result, the errors x1 and x 2 can respectively converge to small regions E1 and E2 around the origin as claimed. Theorem 2 is proven.

(68)

Remark 2. The proposed AGFST control laws (34) and the existing adaptive NTSMC laws in [48] can provide finite-time convergence and reject uncertainties/disturbances without any information on them. They can resolve singularity problem and reduce chattering problem. The proposed control laws (34) include fast power reaching law composed of power reaching 1/2 law Si sgn(Si ) and exponential reaching law Si . When the system states are far away from the IFNTSM surface, the term Si guarantees fast convergence; on the contrary, power reaching

It is easy to verify the following inequality

V̇ + τ1V + τ2V1/2 ≤ 0

Case 2 Suppose that εi < Si(t ) < Sε . As ρi is increasing in accordance with the first equation in (32), the sliding mode manifold is driven to a small region εi around the origin, i.e. Si(t ) ≤ εi . Then ρi shall reduce until ρi <νi . From the second equation in (32), we have

⎧ λ − ⎪ ⎪ 1i ⎨ ⎪λ − 2i ⎪ ⎩

i=1

∑ θ¯2i(

where V0 is the initial value of the introduced Lyapunov function V.

x2i + λ1ix1i + λ2ix1i ri = 0 = φκi ,

The further analysis is discussed from the following two cases. Case 1 Suppose that Si(t ) ≥Sε and ρi ≥νi for all t ≥0. Then, substituting (32) into (65), we obtain

−

(69)

If x1i >ε we have

ρi + ρ¯˜i ρ¯i̇ + + ∑ θ¯1i ρ˜i ⌢

V̇ ≤ − η1Vζ − η2Vζ1/2 −

τ V1/2 + τ2 2 ln 1 0 3τ1 τ2

Case 1

3

−

Lemma 1 in [29], the proposed IFNTSM manifold is established in a finite time tF :

tF ≤

Then, we can obtain

7

is established if the equalities τ1=min(η1,θ1̅ i ) and τ2=min(η2,θ2̅ i ) are satisfied. It is worth noting that for the finite time convergence ρi (t ) must satisfy inequality (56). It means that ρi (t ) shall increase in accordance with (32) until (56) is met that guarantees the positive definiteness of the symmetric matrix Q̅ and validity of (68). After that the finite time tF convergence to the domain Si ≤Sε is guaranteed according to (68). According to

Please cite this article as: Zhang Y, et al. Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.08.012i

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law Si sgn(Si ) provides fast convergence, then the proposed control laws (34) can provide faster convergence speed than standard ST algorithm. The adaptation gains can overcome non-overestimating problem caused by robust control gains, and by equivalent control, high activity of control action is also reduced, then they achieve anti-wasting energy simultaneously. The proposed control laws (34) are second-order sliding mode approach which can provide more accurate control precision than first-order SMC [48]. The differences of the proposed control laws (34) and existing control laws in [48] are t feedback terms ∫ ρ¯i (τ )χi (τ )dτ and uadi(t ). The proposed second0 order control laws (34) can provide superior control performance than the existing first-order control laws in [48] due as asymptotic stability is lost during boundary layer. The existing control laws in [48] need to achieve the insensitivity to bounded disturbances at the price of losing the merits secondorder SMC laws, boundary layer is used to reduce chattering effect of compensation term uadi(t ). Thus, the proposed control laws can achieve better control function than the existing control laws in [48]. In conclusion, the advantages of the proposed AGFST sliding mode control algorithm can be summarized as follows: Firstly, the chattering phenomenon is suppressed effectively. Then, the large overshooting and the high-frequency chattering caused by overestimating the control gains are solved. Finally, the stronger robustness and the better control performance are realized. Above all, the proposed AGFST-RFTSMC approach guarantees the tracking error to converge to a small neighborhood of the origin rapidly and maintains a friendly relationship between low frequency control switching and strong robustness against the unknown uncertainty and actuator fault.

5. Simulation results In this section, the simulations are conducted for demonstrate the performance of the proposed control method and compared with three existing methods. They are adaptive nonsingular terminal sliding mode control (ANTSMC), nonsingular terminal sliding mode control (NTSMC), and feedback linearization (FBL). The control algorithms of ANTSMC and FBL are derived from [48] and [35], respectively. The ANTSMC laws in [48] use the same sliding mode manifold as the proposed manifold in this paper and the conventional NTSM surface is designed as Sn = [Sn1, Sn2, Sn3]T , in which Sni is stated as follows w

Sni = ei + βi ei̇ sgn(ei̇ ),

i = 1, 2, 3

(76)

where βi > 0 and w ∈ (1, 2) are design parameters. Therefore, consider the reentry vehicle system (15). The NTSMC algorithm can be expressed as follows

Mc = G −1 −F − τnS n − σnsig γn(S n) + Ω¨ c

(

)

(77)

where 0 < γn < 1, τn = diag(τn1, τn2, τn3), τni > 0, σn = diag(σn1, σn2, σn3), σni > 0 T γ γ γ sig γn(S n) = ⎡⎣ Sn1 n sgn(Sn1), Sn2 n sgn(Sn2), Sn3 n sgn(Sn3)⎤⎦ , i = 1, 2, 3

The numerical tests in this paper employ a reentry vehicle, whose moments of inertia are Ix = 588791kg⋅m2, Iy = 1303212kg⋅m2, and the products of inertia are Iz = 1534164kg⋅m2, Izx = Ixz = 24242kg⋅m2. In addition, the initial conditions for reentry vehicle are taken as follows, the altitude h = 55.0km , Mach number T T Ma = 9.8, Ω0 = ⎡⎣ 32.0∘ , 2.0∘ , 58.0∘⎤⎦ , ω0 = ⎡⎣ 0.0∘ /s, 0.0∘ /s, 0.0∘ /s⎤⎦ . And T

attitude angle commands are set to be Ωc = ⎣⎡ 30.0∘ , 0.0∘ , 60.0∘⎦⎤ . The planet uncertainties are set in consideration of 10% bias conditions for moments of inertia and products of inertia, 20% bias conditions for aerodynamic coefficients, and 25% bias conditions for atmospheric density. In addition, the external disturbance torque vector takes the form of

⎡ 1 + sin( t )⎤ ⎥ ⎢ ΔM = ⎢ 1 + sin( t )⎥ × 104 N⋅m ⎥ ⎢ ⎢⎣ 1 + sin( t )⎥⎦

(78)

Meanwhile, actuator faults and actuator saturations are conT sidered. The actuator fault scenarios Fδ=⎣⎡ Fδ1,Fδ2,Fδ 3⎤⎦ and the matrix of actuator effectiveness D=diag(δl,δm,δn) are defined as

Fδi = (1 + sin(0.1t )) × 104 N⋅m,

i = 1, 2, 3

⎧ 1, δl = ⎨ ⎩ 0.75 + 0.1 × sin(0.5t + π /3),

if t ≤ 8s if t > 8s

⎧ 1, δm = ⎨ ⎩ 0.5 + 0.1 × sin(0.5t + 2π /3), ⎧ 1, δn = ⎨ ⎩ 0.35 + 0.1 × sin(0.5t + π ),

(79)

(80)

if t ≤ 8s if t > 8s

(82)

if t ≤ 8s if t > 8s

(84)

To validate the robustness and the chattering reduction of the proposed methods, numerical simulations of ANTSMC, NTSMC, FBL, and AGFST-RFTSMC are presented. The total simulation time is 15 s, and the integration step is specified as 0.01 s. In FBL, the control parameters are selected as k p = kd = 3. In addition, the control parameters of the sliding mode controllers are specified in Table 1. The variations of the attitude angles including AOA, sideslip angle and bank angle under ANTSMC, NTSMC, FBL, and AGFST-RFTSMC are shown in Fig. 2. It is obvious that SMC has significant robustness performance in the presence of uncertainties and disturbances. Based on Fig. 3, since FBL relies on the knowledge of the exact model dynamics, the tracking errors under FBL does not converge to zero. Compared with those existing SMC laws, AGFST-RFTSMC achieves the goal of tracking with higher accuracy and faster convergence speed. The sliding surface responses via ANTSMC and AGFST-RFTSMC are shown in Fig. 4. It is obvious that the sliding surface under NTSMC do

Table 1 Sliding mode control parameters. Parameter/Controller Sliding surface parameters Control parameters

NTSMC (i ¼ 1,2,3)

ANTSMC (i ¼ 1,2,3)

βi = 2

λ1i = 3

w = 1.6

ri = 7/9 εi = 0.001

ri = 7/9 εi = 0.001

τn = diag(5, 5, 5)

τ = diag(10, 10, 10) ⌢ ⌢ θi (0) = ϑi(0) = 0.1

α0i = 15 νi = 0.001 δi = 0.5

μ1 = 1.2

ε1i = ε2i = 0.75 p¯i = q¯i = 10

γi = 0.3

μ2 = 0.6

σn = diag(10, 10, 10) γn = 0.8

λ2i = 2

AGFST-RFTSMC (i ¼ 1,2,3)

λ1i = 3

λ2i = 2

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Fig. 2. Comparison of attitude angles via AGFST-RFTSMC, ANTSMC, NTSMC and FBL.

Fig. 3. Comparison of tracking errors for attitude angles via AGFST-RFTSMC, ANTSMC, NTSMC and FBL.

not converge to zero because the adaptive law is not introduced. The system trajectories under ANTSMC and AGFST-RFTSMC move apart from zero at the beginning region although they remain to be zero. However, the sliding surface under AGFST-RFTSMC provides better convergence performance and converges to zero more precisely. From Fig. 5, attitude angular rates have stable convergence characteristic when the vehicle suffers from unknown actuator faults by using the proposed AGFST-RFTSMC scheme. Additionally, the produced command control torques are showed in Fig. 6.

6. Conclusions The main contribution in this paper is to propose a RFTSMC method for the problem of tracking control for a RLV with

unknown uncertainties and actuator faults. First, based on the classical FTSM and NTSM manifold, an IFNTSM manifold is designed to avoid singularity problem and chattering phenomenon. Besides, the proposed manifold can provide fast convergence rate. Second, the ST algorithm combining with adaptive technique and fast reaching law for the RLV is proposed to the final synthesis design. By using dynamically adapted control gains, the proposed AGFST algorithm provides faster convergence rate than the classical ST algorithm and has strong robustness against unknown uncertainties/disturbances and actuator faults growing in time or together with the tracking errors. The key-point of the adaptive gains is to overcome the drawback of overestimating without any knowledge of the bounds of uncertainties/disturbances and actuator faults in advance and resolve robust gains being selected difficult. Simulation results demonstrate that the proposed

Please cite this article as: Zhang Y, et al. Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.08.012i

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Fig. 4. Comparison of sliding surface responses via AGFST-RFTSMC, ANTSMC, NTSMC and FBL.

Fig. 5. Comparison of attitude angular rate via AGFST-RFTSMC, ANTSMC, NTSMC and FBL.

Fig. 6. Comparison of command control torque via AGFST-RFTSMC, ANTSMC, NTSMC and FBL.

RFTSMC system provides rapid and stable attitude control performance, which shows the effectiveness and superiority of the proposed control scheme.

Acknowledgments This work has been supported in part by the National Natural Science Foundation of China (Nos. 11202024, 11572036).

Please cite this article as: Zhang Y, et al. Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.08.012i

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Yao Zhang, Ph.D. student of Beijing Institute of Technology, major in flight vehicle design, flight mechanics and control.

Shengjing Tang, Ph.D., professor, received his Ph.D. degree from Technical University of Munich in 2002, his current research areas are flight mechanics and control, flight vehicle design.

Jie Guo, Ph.D., received his Ph.D. degree from Beijing Institute of Technology in 2010, his current research area is flight mechanics and control.

Please cite this article as: Zhang Y, et al. Adaptive-gain fast super-twisting sliding mode fault tolerant control for a reusable launch vehicle in reentry phase. ISA Transactions (2017), http://dx.doi.org/10.1016/j.isatra.2017.08.012i