Finite time positioning control for a stratospheric airship

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ScienceDirect Advances in Space Research xxx (2019) xxx–xxx www.elsevier.com/locate/asr

Finite time positioning control for a stratospheric airship Yueneng Yang ⇑ College of Aerospace Science and Engineering, National University of Defense Technology, No. 47 Yanwachi Street, Kaifu District, Changsha 410073, China Received 11 September 2018; received in revised form 25 November 2018; accepted 28 December 2018

Abstract The stratospheric airship provides a unique and promising aerostatic platform for broad applications, which requires fast and robust positioning control to support these tasks. A finite time control scheme is proposed to address the problem of positioning control for stratospheric airships subject to dynamics uncertainty. A nonsingular terminal sliding mode controller is designed for positioning control, which overcomes the problem of asymptotical convergence of sliding mode control and the singularity problem of terminal sliding mode control. Under the framework of nonsingular terminal sliding mode control, a fuzzy logic system is employed to approximate the uncertain dynamics of the stratospheric airship, and the fuzzy logic system approximation-based finite time sliding mode controller is designed. The finite-time convergence of positioning errors and the stability of the closed loop system are guaranteed by Lyapunov theory. Finally, the effectiveness and performances of the proposed controller are demonstrated through experimental simulations. Contrasting simulation results illustrate that the proposed controller decreases chattering effectively and ensures faster convergence compared to sliding mode controller. Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.

Keywords: Positioning control; Finite time convergence; Dynamics uncertainty; Fuzzy logic system; Approximation; Stratospheric airships

1. Introduction The stratospheric airship, a typical lighter-than-air vehicle, is considered as a promising platform for various missions (Young et al., 2009; Mueller et al., 2009; Zheng et al., 2015), such as communication relay, reconnaissance, aerial photography, earth observation (Chu et al., 2007; Chaugule and Rajkumar, 2011; Schafer and Reimund, 2008; Yang et al., 2012a), and so on. To guarantee these missions operate successfully, positioning control is highly desirable. In practical situations, the airship dynamics are highly non-linear, with large couplings and uncertainties, bringing on a major difficult of control design.

⇑ Address: College of Aerospace Science and Engineering, National University of Defense Technology, Sany Road, KaiFu District, Changsha 410073, China. E-mail address: [email protected].

To address the problem of positioning control for stratospheric airships, a large variety of control approaches have been proposed in literatures, such as PID control (Zwaan et al., 2000), conventional loop-shaping approach (Schmidt et al., 2007), backstepping design technique (Azinheira and Moutinho, 2006; Hygounenc and Soueres, 2002), sliding mode control (SMC) (Benjovengo, 2009; Yang, 2014), and so on. Among these control approaches, SMC provides an effective and promising approach for positioning, which has a particular property of insensitivity to parametric variations and external disturbance (Ma and Sun, 2016; Li et al., 2018; Guo et al., 2017). Benjovengo (2009) applied SMC to develop the control law for unmanned airships, covering the full flight envelope from hovering to aerodynamic flight. Yang (2014) designed a positioning controller for an autonomous airship using SMC. However, SMC can only guarantee asymptotic stability of the closed loop system due to the use of linear slid-

https://doi.org/10.1016/j.asr.2018.12.038 0273-1177/Ó 2019 COSPAR. Published by Elsevier Ltd. All rights reserved.

Please cite this article as: Y. Yang, Finite time positioning control for a stratospheric airship, Advances in Space Research, https://doi.org/ 10.1016/j.asr.2018.12.038

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Y. Yang / Advances in Space Research xxx (2019) xxx–xxx

ing surface. Therefore it has an obvious drawback, i.e., asymptotical convergence of the closed loop system (Li et al., 2018). Fortunately, terminal sliding mode control (TSMC) has been developed to guarantee finite time convergence by use of nonlinear sliding surfaces (You et al., 2018; Tarek et al., 2017; Sun et al., 2014), which has several advantages such as fast response, finite time convergence and high precision compared to SMC. However, TSMC has an intrinsic drawback called ‘‘singularity” problem, which hinders practical use for nonlinear control systems (Feng et al., 2002; Yang, 2018; Han et al., 2014; Yang and Yan, 2016a). Thus, it is quite necessary to develop a nonsingular method to avoid the singularity problem. Another challenging problem is that the stratospheric airship dynamics are always uncertain in practice. Therefore, the online approximator or estimator is required to assure the control performance. Fortunately, the fuzzy logic system (FLS) with excellent approximation capability represents an attractive alternative (Zhai et al., 2016; Cheng et al., 2016; Yang et al., 2013). In this paper, a finite time sliding mode control (FTSMC) approach, i.e., fuzzy logic system approximation-based finite time sliding mode control (FLS-FTSMC) approach, is proposed for positioning control of stratospheric airships subject to dynamics uncertainty. A nonsingular terminal sliding manifold (NTSM) is presented to design the nonsingular terminal sliding mode controller (NTSMC). Under the framework of FTSMC, a FLS is developed to online approximate the uncertain dynamics of the stratospheric airship. Thus, the proposed FLS-FTSMC, not only obtains the property of finite time convergence, but also addresses the problem of dynamics uncertainty. The main contributions of this work can be summarized as follows. (1) The FTSMC addresses the problem of asymptotical convergence of SMC. (2) A FLS is developed to online approximate the uncertain dynamics of the stratospheric airship, which alleviated the difficulty of system modeling in practice. (3) The finite time convergence and stability of the closed loop system is guaranteed. The rest of this paper is organized as follows. Section 2 formulates the problem of positioning control. Section 3 designs the FLS-FTSMC. Section 4 conducts the experimental simulations to verify the performance of FLSFTSMC. Finally, conclusions are given in Section 5.

Desired position

xb r

ob yb

oE

yE

Fig. 1. Sketch map of positioning control of a stratospheric airship.

g_ ¼ J ðgÞv

ð1Þ

M v_ þ N þ G ¼ s

ð2Þ

where g ¼ ½x; y; w , x, y and w are the x-coordinate, ycoordinate and orientation in the earth-fixed frame T oE xE y E , respectively; v ¼ ½u; v; r , u, v and r are the forward speed, lateral speed, and yaw angular velocity of heading in the body-fixed frame ob xb y b , respectively; s ¼ ½su ; sv ; sr T , su , sv and sr are the forward control forces, lateral control forces and heading moment, respectively; JðgÞ denotes the transformation matrix from the body fixed frame to the earth fixed frame; M, N and G are the inertial matrix, the vector function of nonlinear dynamics terms and the vector function of buoyancy and gravitational forces and torques (Yang et al., 2012b), respectively, and the vector function of buoyancy and gravitational forces and torques can be approximated as zero if the stratospheric airship is with neutral buoyancy. The expression of M, N and J ðgÞ are given as follows: 2 3 m þ mu 0 my G 6 7 M¼4 0 ð3Þ m þ mv mxG 5 0 mxG Iz þ Ir 2 3 cosw sinw 0 6 7 J ðgÞ ¼ 4 sinw cosw 0 5 ð4Þ T

0

0

1

T

N ¼ ½N u ; N v ; N r 

ð5Þ

where N u ¼ ðm þ m22 Þvr þ mxG r2 þ QK2=3 ðC X cosacosb þ C Y cosasinbÞ

2. Modeling and formulation The kinematics and dynamics of a stratospheric airship with neutral buoyancy can be simplified as horizontal motion with three-degree-of-freedom, as show in Fig. 1, which can be described as follows (Zheng et al., 2017; Chen et al., 2016; Wang et al., 2018; Chen et al., 2014):

N v ¼ ðm þ m11 Þur þ my G r

ð6Þ

2

þ QK2=3 ðC X sinb þ C Y cosbÞ N r ¼ my G vr  mxG ur þ QKC n

ð7Þ ð8Þ

where K is the volume of the airship, Q is the dynamic pressure, m is the airship mass, xG , y G and zG are the distance

Please cite this article as: Y. Yang, Finite time positioning control for a stratospheric airship, Advances in Space Research, https://doi.org/ 10.1016/j.asr.2018.12.038

Y. Yang / Advances in Space Research xxx (2019) xxx–xxx

from center of gravity to center of volume of the airship, mu , mv and I r are the added mass and inertia, and C X , C Y and C n are the aerodynamics coefficients. The control problem of the current paper is formulated as follows: given any initial states g0 and v0 , and the desired position and orientation gd , consider the kinematics and dynamics of the stratospheric airship given by (1) and (2), respectively, design an appropriate law such that the T position and orientation errors ge ¼ ½xe ; y e ; we  converges to zero in finite time, i.e., lim k ge k ¼ lim k g  gd k ¼ 0

t!tf

t!tf

ð9Þ

3

The differential of the candidate Lyapunov function given by (14) is less than zero if q and c are both diagonal positive definite matrices. The differential of the candidate Lyapunov function given by (14) equals zero once the sliding surface given by (11) equals zero. Then, the following equation can be derived lim s ¼ lim ðe_ þ aeÞ ¼ lim ðaðg  gd Þ þ e_ Þ ¼ 0

t!1

t!1

t!1

ð16Þ

Considering a is a diagonal positive definite matrix, the following equation is derived from (16) lim g ¼ gd ;

t!1

lim e_ ¼ 0

ð17Þ

t!1

where tf denotes the time taken to approach the desired position and orientation from the initial state. Remark 1. The SMC designed as (13) ensures Lyapunov stability of the nonlinear dynamics system given by (1) and (2), and the positioning error converges to zero asymptotically.

3. Positioning control design 3.1. Sliding mode control The control errors between the actual position and orientation and the desired position and orientation are defined as e ¼ g  gd

ð10Þ

The linear hyperplane-based sliding surface is defined as (Itkis, 1976; Utkin, 1992; Zinober, 1993) s ¼ e_ þ ae

ð11Þ

where a is a diagonal positive definite matrix. Based on the sliding surface given by (11), the corresponding approaching law is selected as (Itkis, 1976; Utkin, 1992; Zinober, 1993) s_ ¼ qs  c signðsÞ

ð12Þ

where signðÞ denotes the sign function, and q and c are both diagonal positive definite matrices. Then, the SMC is designed as (Yang et al., 2014; Yang and Yan, 2018) s ¼ Mv þ N þ G  qs  c signðsÞ

ð13Þ

Theorem 1. Given the airship kinematics and dynamics as (1) and (2), respectively, if the SMC is designed as (13), then the sliding surface given by (11) will be reached asymptotically and the control errors will converge to zero asymptotically. Proof. A candidate Lyapunov function is chosen as follows 1 V 1 ¼ sT s 2

ð14Þ

Differentiating (14) with respect to time, and using (12) yields V_ 1 ¼ sT s_ ¼ sT ðqs  c signðsÞÞ ¼ sT qs  ck s k

ð15Þ

3.2. Nonsingular terminal sliding mode control The NTSM based on the nonlinear combination of control errors and its derivative is designed as (Yang and Yan, 2016a) s ¼ e þ k1 e_ p=q

h

iT

ð18Þ

p=q p=q p=q , k ¼ diagðk1 ; k2 ; where 1 < p=q < 2, e_ p=q ¼ e_ 1 ; e_ 2 ; e_ 3

k3 Þ, ki > 0 is a designed parameter, and i ¼ 1; 2; 3. The time derivative of NTSM is   p s_ ¼ e_ þ k1 diag e_ p=q1 €e q

ð19Þ

Differentiating (18) and using (2) yields   €e ¼ s þ I 33  M g €g  N g g_  G g

ð20Þ

where I 33 is the identity matrix with three dimension. Denote   f ¼ I 33  M g €g  N g g_  G g ð21Þ and (20) is rewritten as €e ¼ s þ f

ð22Þ

Based on the NTSM given by (18), the NTSMC is designed as follows: q ð23Þ s ¼ f  k_e2p=q  k signðsÞ p where k ¼ diagðk 1 ; k 2 ; k 3 Þ, k i > 0 is a designed parameter, and i ¼ 1; 2; 3. Theorem 2. Given the airship kinematics and dynamics as (1) and (2), respectively, if the NTSMC is designed as (23), then the NTSM given by (18) will be reached in finite time and the control errors will converge to zero in finite time.

Please cite this article as: Y. Yang, Finite time positioning control for a stratospheric airship, Advances in Space Research, https://doi.org/ 10.1016/j.asr.2018.12.038

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Y. Yang / Advances in Space Research xxx (2019) xxx–xxx

Proof. Another candidate Lyapunov function is chosen as

The detailed design of FLS-FTSMC is given step by step as follows.

1 V 2 ¼ sT s 2

Algorithm 1: FLS-FTSMC

ð24Þ

Differentiating (24) with respect to time, and using (18) and (21) yields     V_ 2 ¼ sT e_ þ pq k1 diag e_ p=q1 €e n o   ¼ sT e_ þ pq k1 diag e_ p=q1  qp k_e2p=q  k signðsÞ n o   ¼ sT  pq k1 k diag e_ p=q1 signðsÞ 6 j k s k < 0

ðs–0Þ 

where j ¼ pq k k1 k diag e_

 p=q1

ð25Þ k > 0.

Remark 2. The in-equation (25) illustrates that the condition of Lyapunov stability is satisfied, and the finite-time approach of NTSM given by (18) can be guaranteed. The differential of the candidate Lyapunov function given by (24) equals zero once the sliding surface given by (11) equals zero. Therefore, the following equation is derived     lim s ¼ lim e þ k_ep=q ¼ lim ðg  gd Þ þ k_ep=q ¼ 0 ð26Þ t!tf

t!tf

t!tf

lim g ¼ gd ; lim e_ ¼ 0

t!tf

Input: 1) Desired position and orientation gd ; 2) Actual velocities v and position and orientation g. Output: Positioning control input. Step 1: Design of the sliding manifold: a) Compute the positioning error e ¼ g  gd ; b) Design the nonsingular terminal sliding manifold s ¼ e þ k1 e_ p=q ; c) Select the gain k; Step 2: Design of the control law: a) Design the FTSMC s ¼ f  qp k_e2p=q  k signðsÞ; b) Design the FLS approximatior f^ðxÞ; c) Design the FLS-FTSMC s ¼ f^  qp e_ k2p=q  k signðsÞ; Step 3: Stability proof of the closed-loop system a) Select a Lyapunov function candidate V; b) Compute the differential of V; c) Check the sign of the differential of V; d) Analyze the convergence of positioning error e. Step 4: Termination If the tolerance of control error is satisfied, terminate the algorithm and output s. Otherwise, go to step 1.

t!tf

ð27Þ

where tf = tr + ts, tr is the time when s reaches zero, and ts is the finite time which is expressed as (Feng et al., 2002; Han et al., 2014) Z 0 dei pkq=p 1q=p i ts ¼ kiq=p ½ei ðtr Þ ð28Þ ¼ q=p ðtÞ p  q ei ðtr Þ ei where ei 2 e, and i ¼ 1; 2; 3. Remark 3. It is proven that the NTSMC designed as (23) guarantees the control errors converge to zero in finite time. 3.3. Nonsingular terminal sliding mode control based on fuzzy approximation It has been proven that the NTSMC designed as (23) can guarantee finite time convergence of the control errors and provides a potential approach for positioning control of stratospheric airships. Unfortunately, the nonlinear terms of airship dynamics given by (21) are usually uncertain in practice, and thus the designed controller (23) can’t be accomplished in practice operation. To solve this problem, a FLS approximator is designed to online approximate the uncertain terms of the airships dynamics (PedroNeto, 2015; Yang and Yan, 2016b Qi et al., 2017). The FLS is brought into the framework of NTSMC to construct the FLS-FTSMC, as shown in Fig. 2.

Lemma 1 ((PedroNeto, 2015; Qi et al., 2017)). Let gðxÞ be a continuous vector function defined on a compact Xx 2 Rn , for any constant e > 0, there exists a FLS g^ðxjhÞ such that sup k g ðxÞ  g^ðxjhÞ k < e

ð29Þ

x2Xx

By Lemma 1, it implies that FLS can be used as a universal approximator to approximate the nonlinear terms given by (21) with a very small error. Based on the FLS approximator given by (29), the FLSFTSMC is designed as q s ¼ f^  k_e2p=q  k signðsÞ ð30Þ p where f^ ¼ HT ff ðxÞ is the approximation of the nonlinear h iT  T terms f,x ¼ eT ; e_ T ; gT ; g_ T ; €gT , H ¼ hTf 1 ; hTf 2 ; hTf 3 is the weighting parameters of the approximatior, h iT ff ðxÞ ¼ nTf 1 ðxÞ; nTf 2 ðxÞ; nTf 3 ðxÞ is the basic function vector, where hf i ; ði ¼ 1; 2; 3Þ are the optional parameters, n P l l ðxj Þ  1 T j¼1 F j m l , nðxÞ ¼ n ðxÞ; . . . ; n ðxÞ , n ðxÞ ¼ Pm  n P lF l ðxj Þ l¼1 j¼1 j

l ¼ 1; 2 . . . ; m, where lF lj ðxj Þ is the degree of membership function (Yang and Yan, 2016b; Qi et al., 2017).

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Y. Yang / Advances in Space Research xxx (2019) xxx–xxx

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Fig. 2. Block diagram of FLS-FTSMC.

The architecture of the FLS approximator is given as follows (see Fig. 3). Define the optimal value of the weighting parameters H as follows (Meguenni, 2012; Ullah et al., 2015)  

^    H ¼ arg min sup f ðxjH Þ  f ðxÞ ð31Þ 

 T   1 1 H V 3 ¼ sT s þ tr H c1 W 2 2

ð34Þ



ð32Þ

where H ¼ H  H. Differentiating (34) and using (19), (30) and (32) yields  T  _ V_ 3 ¼ sT s_  tr H c1 H W 

T   p ¼ sT  k1 k diag e_ p=q1 signðsÞ þ H hðxÞ þ eðxÞ q  T  _  tr H c1 H

where em is the least approximating error, and it is bounded.

ð35Þ

H

x2Xx

According to the approximation properties of FLS (Meguenni, 2012; Ullah et al., 2015), there exist an optimal weighting parameters H such that f ¼ HT ff ðxÞ þ em

Theorem 3. Given the airship kinematics and dynamics by (1) and (2), respectively, if the FLS-FTSMC is designed as (30), in which the nonlinear terms of airship dynamics defined by (21) is approximated by FLS and the adaptation law is designed as (33), then the NTSM given by (18) will be reached in finite time and the positioning errors will converge to zero in finite time. The adaptation law is designed as _ ¼ cW hðxÞsT  bW k s kcW H H

ð33Þ

where bW > 0, and cW is a diagonal positive definite matrix. Proof. Choose the following Lyapunov function candidate

W

where eðxÞ is the approximating error. Substituting the adaptation law (33) into (35) yields   p V_ 3 ¼  sT k1 k diag e_ p=q1 signðsÞ q  T   p=q1  p T 1 s k diag e_ þ ð36Þ e þ bW k s ktr H H q According to the property of the Frobenius norm, it is obtained  T     2 bW k s ktr H H 6 bW k s k k H kF k H kF  k W kF    2 6 bW k s k Hmax k H kF  k H kF   2 2 þ bW Hmax ksk 6 bW k s k k H kF  Hmax 2 4 6 bW Hmax ksk 4 2

ð37Þ where Hmax is the maximum value of the Frobenius norm of H . Denote   p ð38Þ n ¼ k k1 diag e_ p=q1 kk e k q Fig. 3. Architecture of the FLS approximator.

f ¼ bW

H2max 4

ð39Þ

Please cite this article as: Y. Yang, Finite time positioning control for a stratospheric airship, Advances in Space Research, https://doi.org/ 10.1016/j.asr.2018.12.038

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Y. Yang / Advances in Space Research xxx (2019) xxx–xxx

and (36) can be expressed as

Table 1 Model parameters of the stratospheric airship. Model parameters

Value

m mu mv Iz Ir xc yc zc K

9.07  103 kg 1.12  103 kg 7.24  103 kg 18.76  108 kg m2 9.1 kg m2 0m 0m 10 m 1.03  105 m3

V_ 3 6 ðj  n  fÞ k s k

ð40Þ

If the gains j, n and f satisfy the following in-equation jP nþf

ð41Þ

then V_ 2 < 0 can be guaranteed. Remark 4. It is concluded that all the signals of the closed loop system are bounded under the condition given by (41). It is proven that the FLS-FTSMC designed as (30) assures the control errors converge to zero in finite time.

1000 900 End position

800 700

y(m)

600 500 400 300 200 Initial position

100 0 0

500

1000

1500

2000

x(m)

(a) Positioning process under SMC 10

τu(N)

x (m) e

2000

1000

0 0

50

100

150

200

250

5 0 -5 0

300

50

100

500

20

τv(N)

40

e

y (m)

1000

0 -500 0

50

100

150

200

250

-20 0

300

250

300

50

100

150

200

250

300

200

250

300

Time(s)

2

50

τr(Nm)

ψe(rad)

200

0

Time(s)

1 0 -1 0

150

Time(s)

Time(s)

50

100

150

200

250

300

0 -50 0

50

100

150

Time(s)

Time(s)

(b) Positioning errors under SMC

(c) Control inputs of SMC

Fig. 4. Simulation results of positioning control using SMC.

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Y. Yang / Advances in Space Research xxx (2019) xxx–xxx

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following desired position T gd ¼ ½1800 m; 950 m; 0:86 rad .

4. Experimental simulations In the current section, experimental simulations have been conducted to verify the performance of the proposed control approach. The main parameters of the stratospheric airship are available in Table 1, and it is assumed that all the parameters are uncertain with a random error of the order of 15% and the disturbance in lateral direction is sd ¼ 0:2cosðp=100tÞ N. The initial position, orientation and velocities of the stratospheric airship are given by g0 ¼ ½150 m; 50 m; T T 0:001 rad andv0 ¼ ½8 m=s; 3 m=s; 0 rad=s , and the stratospheric airship is required to approach to the

and

orientation

4.1. Simulation results of SMC To evaluate the performance of the proposed FLS-FTSMC, the positioning control using SMC is performed as a compared simulation. The designed parameters of SMC are selected as q ¼ diagð1:5; 3; 12Þ, c ¼ diagð15; 15; 15Þ. Simulation results of SMC are shown in the following figures. Fig. 4(a) shows the positioning process under SMC, the airship from an initial position (150 m, 50 m) with the

1000 900

End position

800 700

y(m)

600 500 400 300

Initial 200 position 100 0

0

500

1000

1500

2000

x(m)

2000

6

1000

4

τu(N)

xe(m)

(a) Positioning process under FLS-FTSMC

0 -1000 0

50

100

150

200

250

2 0 0

300

50

100

τv(N)

e

y (m)

500

50

100

150

200

250

300

200

250

300

200

250

300

2 0 0

300

50

100

150

Time(s) 100

τr(N⋅m)

1

ψe(rad)

250

4

Time(s) 0.5 0 -0.5 0

200

6

1000

0 0

150

Time(s)

Time(s)

50

100

150

200

250

300

0 -100 -200 0

50

100

(b) Positioning errors under FLS-FTSMC

150

Time(s)

Time(s)

(c) Control inputs of FLS-FTSMC

Fig. 5. Simulation results of positioning using FLS-FTSMC.

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Y. Yang / Advances in Space Research xxx (2019) xxx–xxx

orientation w0 ¼ 0:001 rad approaches to the desired position (1800 m, 800 m) effectively. As shown in Fig. 4(b), the positioning errors converge to a small neighborhood of zero asymptotically. Fig. 4(c) displays the control inputs of SMC, which switch from one amount to anther discontinuously, and the chattering phenomenon is obvious. 4.2. Simulation results of FLS-FTSMC The Gaussian function is employed to construct the FLS approximator (Meguenni, 2012), and the designed parameters of the FLS-FTSMC are selected as k ¼ diagð2; 4; 2Þ, a ¼ diagð5; 5; 5Þ, p ¼ 5, q ¼ 3. Simulation results of FLS-FTSMC are shown in the following figures. Fig. 5(a) demonstrates the positioning control result under FLS-FTSMC, the airship from an initial point (150 m, 50 m) with the orientation w0 ¼ 0:001 rad approaches to the desired position (1800 m, 950 m) effectively. The positioning errors converge to zero in finite time, as shown in Fig. 5(b), which illustrates the control performance of the designed FLS-FTSMC. In contrast to asymptotical convergence of positioning errors under SMC, the positioning errors under FLS-FTSMC converge to zero in finite time. The control inputs are displayed in Fig. 5(c). Compared to the control inputs of SMC, the chattering has been decreased. Remark 5. The experimental simulations demonstrate that both the SMC and FLS-FTSMC are effective for positioning control of stratospheric airships. Compared to SMC using linear sliding manifold, the FLS-FTSMC assures the finite time convergence of both control errors and sliding manifold. 5. Conclusion In this paper, a FLS-FTSMC is proposed for positioning control of stratospheric airships subject to dynamics uncertainty, in which a NTSM is employed to design the NTSMC to guarantee finite time convergence and a FLS approximator is developed to online approximate the uncertain dynamics of the stratospheric airship. The effectiveness of the designed FLS-FTSMC is verified via the experimental simulations, and compared simulation with SMC demonstrates the advantages of FLS-FTSMC, which ensures finite time convergence of positioning error and approximates the uncertain dynamics of the stratospheric airship. Acknowledgements This work is supported by National Natural Science Foundation of China, China (No. 11502288), Aeronautical Science Foundation of China, China (2017ZA88001) and Scientific Research Project of National University of Defense Technology, China (ZK17-03-32), and the authors also deeply indebted to the editors and reviewers.

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Please cite this article as: Y. Yang, Finite time positioning control for a stratospheric airship, Advances in Space Research, https://doi.org/ 10.1016/j.asr.2018.12.038