Research ArticleNonsingular Fast Terminal Adaptive Neuro-sliding ModeControl for Spacecraft Formation Flying Systems

XiaohanLin 1XiaopingShi 1 ShilunLi 2 SingKiongNguang 3 andLiruoZhang 3

1Control and Simulation Center Harbin Institute of Technology Harbin 150080 China2School of Astronautics Harbin Institute of Technology Harbin 150001 China3Department of Electrical Computer and Software Engineering e University of Auckland Auckland 1142 New Zealand

Correspondence should be addressed to Xiaohan Lin hitlinxhhotmailcom

Received 9 March 2020 Revised 23 April 2020 Accepted 11 May 2020 Published 30 May 2020

Academic Editor Xianming Zhang

Copyright copy 2020 Xiaohan Lin et al is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

In this paper a nonsingular fast terminal adaptive neurosliding mode control for spacecraft formation flying systems is in-vestigated First a supertwisting disturbance observer is employed to estimate external disturbances in the system Second a fastnonsingular terminal sliding mode control law is proposed to guarantee the tracking errors of the spacecraft formation convergeto zero in finite time ird for the unknown parts in the spacecraft formation flying dynamics we proposed an adaptiveneurosliding mode control law to compensate them e proposed sliding mode control laws not only achieve the formation butalso alleviate the effect of the chattering Finally simulations are used to demonstrate the effectiveness of the proposedcontrol laws

1 Introduction

With the increasing applications of the spacecraft formationflying system the importance of improving system per-formance of complicated spacecraft with practical controldesign has gained much attention over recent years [1ndash5] Inthe spacecraft formation flying system stability is the mostessential problem to be solved [6] Many control methodshave been applied to solve these problems in the literaturesuch as the adaptive control [7] sliding mode control [8]backstepping technique control [9] fuzzy control [10]optimal control [11] and Hinfin control [12] However alongwith the goal of stabilization for the spacecraft the distur-bances and uncertainties should also be considered which isoften neglected in some research [13]

Sliding mode control is widely used due to its advantagesof robustness against external disturbances [14ndash16] andparameter uncertainties [17] especially in spacecraft for-mation flying system In sliding mode control chattering is acommon phenomenon that often degrades system perfor-mance and even causes instability To deal with this problemthe terminal sliding mode control is developed in which

some nonlinear terms are introduced in the sliding surfaceIn [18] the case of spacecraft final approach combining withthe continuously differential collision avoidance state con-straint function is studied and an improved terminal slidingsurface is presented In [19] based on the conditions ofexternal disturbances and inertial uncertain parameters Ranet al have proposed the adaptive fuzzy terminal slidingmode control for the second-order spacecraft attitude ma-noeuvre system and the convergence time of the slidingsurface has been reduced In [20] a new nonsingular ter-minal sliding mode (NTSM) surface has been introduced toeliminate singularity within the finite time An adaptivenonsingular terminal sliding mode control also has beenpresented for an attitude tracking of spacecraft with actuatorfaults to avoid singularity [21] Although several drawbacksof the sliding mode control have been compensated by themodified control laws due to the high-frequency reachingcontrol term in the control input [22] the chatteringproblem still needs to be solved In [23] a fractional ordercontrol law based on the nonsingular terminal supertwistingsliding mode control is proposed to achieve system stabilityand accurately estimate the unknown model Consequently

HindawiComplexityVolume 2020 Article ID 5875191 15 pageshttpsdoiorg10115520205875191

the nonsingular fast terminal sliding mode surface has theability to reduce the chattering and solve the singularityproblem

In practical spacecraft formation flying systems internaland external disturbances are inevitable Many controlmethods have been used to solve this problem [24 25] edisturbance observers have been widely considered to es-timate the disturbances [26] In [27] an extended stateobserver is introduced to estimate disturbances for themultiagent systems with unknown nonlinear dynamics anddisturbances Furthermore by simultaneously taking theoutput measurement and delayed control input into ac-count the extended observer has been used to estimate theunmeasurable system states and the additive disturbances[28] To have a smaller estimation error of the disturbancehigh-order disturbance observers have been investigated in[29] Among all types of high-order observer the super-twisting disturbance observer stands out due to its ability ofavoiding the excessively high order observer gains that cancause sensor noise within fixed bandwidth [30]

Despite the external disturbances uncertainties in prac-tical formation flying systems are also inevitable howevertheir impacts on the systemrsquos performance have not beenstudied in most of the aforementioned results In practiceparameter uncertainties may have direct impact on systemperformance [31] hence it is critical to deal with the problemof uncertainties Due to the advantage of the neural networksin approximating nonlinear functions it can be applied indealing with the nonlinearity in the spacecraft formationsystems as well [32ndash34] Based on neural network andbackstepping techniques a distributed adaptive coordinatedcontrol algorithm has been proposed in [35] which effectivelysolves the chattering problem To achieve robustness in [36] arobust adaptive fuzzy PID-type sliding mode control has beendesigned with the neural network with which the systemstability is ensured and the transient performance is improvedIn [37] the global adaptive neural control for a general class ofnonlinear robot manipulators has been proposed in which thetransient performance and robustness are both improvedCompared with other neural networks [38ndash40] the RadialBasis Function (RBF) neural network stands out given itsstrong robustness and excellent performance of nonlinearapproximation [41] erefore it is desirable to use the RBFneural network to estimate and compensate for the nonlinearfunctions in the spacecraft formation flying system

Motivated by the above observations in this paper anovel nonsingular fast terminal adaptive neurosliding modecontrol for the spacecraft formation flying system is pro-posede proposed control law guarantees that the trackingerrors converge to zero in finite time By using an adaptiveRBF neural network the uncertainties in the spacecraftformation flying system is compensated e contributionsof the paper can be summarized as follows

(1) External disturbances are estimated by a super-twisting disturbance observer in finite time esupertwisting disturbance observer can avoid highobserver gains that may amplify the noises of thesensors

(2) Compared with [42] our novel fast nonsingularterminal sliding mode control law achieves thestability of spacecraft formation flying system withinfinite time Moreover the singularity is avoided andthe chattering problem is also alleviated

(3) If some of the nonlinear dynamics in the spacecraftformation system are unknown an adaptive RBFneural network control law is proposed to approx-imate those nonlinear dynamics

e remaining parts of the paper are arranged as followse background of the spacecraft formation flying system isintroduced in Section 2 In Section 3 the proposed controllaws and stability analysis are presented Simulation resultsare given to verify the effectiveness of the proposed controllaws in Section 4 Finally some conclusions are given inSection 5

11 Notations e following notations are used throughoutthis paper For a vector x [x1 x2 xn]T isin Rn and|x|d ≜ diag |x1| |x2| |xn|1113864 1113865 the standard sign function isdefined as sign(x) [sign(x1) sign(x2) sign(xn)]Te Euclidean norm of vector x is denoted as x

xTx

radic

For a matrix A isin Rntimesn A1 denotes the maximum absolutecolumn sum norm A2

λmax(ATA)

1113968is the 2-norm

where λmax means the maximum eigenvalue tr(A) denotesthe trace of A Intimesn denotes n times n identity matrix For a three-dimensional vector x [x1 x2 x3]

T the skew-symmetricmatrix xtimes isin R3times3 is defined as follows

xtimes

0 minus x3 x2

x3 0 minus x1

minus x2 x1 0

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦ (1)

2 Spacecraft Formation Systemsand Preliminaries

In this section the spacecraft formation flying system andsome useful lemmas are presented

21 Spacecraft Formation Flying System e relative orbitmotion under the chaser spacecraft coordinate frame isshown in Figure 1 FI denotes the equatorial inertialcoordinate frame (OIXIYIZI) Ft is located at the cen-troid of the target spacecraft and the coordinate axis ofthe target spacecraft coincides with the main inertia axisFI the xt-axis points to the solar panel the yt-axis isperpendicular to the longitudinal plane of xt-axis andzt-axis satisfies the rule of the right hand e chaserspacecraft coordinate frame Fc is similar to the coordi-nate frame Ft

Assume that the active control force of the targetspacecraft is zero and the dynamic models of the targetspacecraft and the chaser spacecraft are given as follows

2 Complexity

eurort minusμe

r3trt +

dct

mt

(2)

eurorc minusμe

r3crc +

dcc

mc

+fcc

mc

(3)

where mt and mc denote the masses of the target spacecraftand the chaser spacecraft and dct isin R3 and dcc isin R3 denotethe external perturbation forces of the target spacecraft andthe chaser spacecraft respectively fcc isin R3 is the controlforce of the chaser spacecraft rt rt2 and rc rc2 wherert and rc denote the position vectors of the target spacecraftand chaser spacecraft to the geocentric respectively

Define ρ rc minus rt then from (2) and (3) the relativeacceleration vector is obtained as

euroρ minusμe

r3crc +

μe

r3trt +

dcc

mc

minusdct

mt

+fcc

mc

(4)

Furthermore let ρc denote the projection vector in thechaser spacecraft body coordinate frame Fc which denotes

fc minusμe

r3crc +

μe

r3trt

dc dcc minusmcdct

mt

uc fcc

(5)

en

euroρc + ωtimesc ωtimes

c ρc( 1113857 + 2ωtimesc _ρc + _ωtimes

c ρc fc +dc

mc

+uc

mc

(6)

where ωc [ωc1ωc2ωc3]T is the angular velocity of Fc

relative to the inertial coordinate frame FI under Fc frameIn this paper the desired relative position and velocity

under Fc frame are assumed to be ρcd and 0 respectivelyen the error vector can be expressed as ec ρc minus ρcd then

(6) can be written as the EulerndashLagrange equation form toexpress the orbit relative motion

mc euroec + Ac _ec + Bcec + gc uc + dc (7)

where

Ac 2mcωtimesc

Bc mc

minus ω2c2 minus ω2

c3 ωc1ωc2 minus _ωc3 ωc1ωc3 + _ωc2

ωc1ωc2 + _ωc3 minus ω2c1 minus ω2

c3 ωc2ωc3 minus _ωc1

ωc1ωc3 minus _ωc2 ωc2ωc3 + _ωc1 minus ω2c1 minus ω2

c2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

gc minus mcfc + mc eurordc + Ac _rdc + Bcrdc

(8)

e Modified Rodrigues Parameters (MRPs) are used todescribe the attitude dynamics of the chaser spacecraft whichcan be defined as

σ 1113954q

1 + q ke tan

φ4

(9)

where q [q 1113954qT]T denotes attitude quaternion and ke andφ isin (minus 2π 2π) denote Euler axis and Euler angular of theattitude rotation respectively

en the attitude kinematics and the dynamics of thechaser spacecraft are given as

_σ G(σ)ωc

G(σ) 14

1 minus σTσ1113872 1113873I3times3 + 2σtimes+ 2σσT

1113960 1113961

J _ωc minus ωtimesc Jωc + uτ + dτ

(10)

where J isin R3times3 denotes the inertia matrix of the chaserspacecraft and uτ isin R3 and dτ isin R3 denote the controltorque and the external disturbance torque of the chaserspacecraft respectively

Defining the error attitude σe and the error angularvelocity ωe as

OI

ZI

YI

XI

ycc

xcc

zcc

yct

xct

zct

Earth plane

Solar arrayChaser spacecra

rc

rt

Target spacecra

Solar array

Docking port o ct

Docking port o cc

Figure 1 e relative orbit motion model of the spacecraft formation flying system

Complexity 3

σe 1 minus σTdσd( 1113857σ minus 1 minus σTσ( 1113857σd + 2σtimesσd

1 + σTσσTdσd + 2σTdσ

ωe ωc minus R σe( 1113857ωd

R(σ) I3times3 minus4 1 minus σTσ( 1113857

1 + σTσ( )2 σ

times+

8σtimes

1 + σTσ( )2 σ

times

(11)

where σd isin R3 denotes the desired attitude and ωd isin R3

denotes the desired angular velocity Furthermore the at-titude model of the chaser spacecraft can be expressed as theEulerndashLagrange form

Mτeuroσe + Aτ _σe + gτ Gminus T σe( 1113857dτ + G

minus T σe( 1113857uτ (12)

where

Mτ Gminus T σe( 1113857JG

minus 1 σe( 1113857

G σe( 1113857 14

1 minus σTe σe1113872 1113873I3times3 + 2σtimese + 2σeσ

Te1113960 1113961

Aτ minus Gminus T σe( 1113857 JG

minus 1 σe( 1113857 _σe1113872 1113873timesG

minus 1 σe( 1113857 + J _Gminus 1 σe( 11138571113876 1113877

+ Gminus T σe( 1113857 J R σe( 1113857ωd( 1113857

times+ R σe( 1113857ωd( 1113857

timesJ1113858

minus JR σe( 1113857ωd( 1113857times1113859G

minus 1 σe( 1113857

gτ Gminus T σe( 1113857 R σe( 1113857ωd( 1113857

timesJR σe( 1113857ωd + JR σe( 1113857 _ωd( 1113857

(13)

In order to describe the attitude-orbit coupling dynamicsmotion of the spacecraft formation uniformly define

xe σe

ec

1113890 1113891

u uτ

uc

1113890 1113891

d dτ

dc

1113890 1113891

(14)

en based on (5) and (22) the attitude-orbit couplingmodel is given as

_xe xde (15)

M _xde minus Axde minus g + C1u + C2d (16)

where

M Mτ 03times3

03times3 mcI3times31113890 1113891 (17)

A Aτ 03times3

03times3 Ac

1113890 1113891 (18)

g gτ

Bcec + gc

1113890 1113891 (19)

C1 C2 Gminus T σe( 1113857 03times3

03times3 I3times31113890 1113891 (20)

Remark 1 It can be seen in (12) the attitude and orbit in thespacecraft formation flying system are coupled erefore itis necessary to build the spacecraft coupling motion model(15) and (16) to enhance the control accuracy and efficiencyFrom (15) and (16) it can be seen that the coupling of at-titude and orbit is reflected by the matrix Bc and the vectordτ simultaneously Bc represents the attitude effect on orbitIn addition the gravity gradient torque contains orbit in-formation and the disturbance torque can reflect the in-fluence of the orbit on the attitude thus dτ can reflect theinfluence of the orbit on the attitude

Assumption 1 e external disturbance vectord [d1 d6] in the spacecraft formation flying system(15) and (16) is assumed to satisfy dledm and |di

|le ςiwhere dm gt 0 and ςi gt 0

Assumption 2 All the state variables of the spacecraft for-mation flying system are assumed to be measurable

22 Preliminary Knowledge In this section some relatedlemmas and definitions in this paper are given below

Lemma 1 (see [43]) For _x f(x t) f(0 t) 0 andx isin U0 sub Rn where f U0 times R+⟶ Rn is continuous withrespect to x on an open neighborhood U0 of the origin x 0Suppose that there are a positive-definite function V(x t)

(defined on 1113954U times R+ where 1113954U isin U0 isin Rn is a neighborhood ofthe origin) real numbers cgt 0 and 0lt αlt 1 such that_V(x t) + cVα(x t) is negative semidefinite on 1113954U en

V(x t) is locally finite-time convergent or equivalently andbecomes 0 locally in finite time with its setting timeTleV(x(t0) t0)

1minus αc(1 minus α) for a given initial conditionx(t0) in a neighborhood of the origin in 1113954U

Lemma 2 (see [44]) Suppose Lyapunov function V(1113954z) isdefined on a neighborhood U sub Rn of the origin and _V(1113954z) +

λ0Vℓ le 0 in which λ0 gt 0 0lt ℓ lt 1 en there exists an areaU0 sub Rn such that any V(1113954z) that start from U0 sub Rn canreach V(1113954z) equiv 0 within finite time Tf leV1minus ℓ(0)λ0(1 minus ℓ) inwhich V(0) is the initial time of V(1113954z)

Definition 1 (see [45]) For the autonomous system_1113954z f(1113954z) (21)

where 1113954z isin RN and f(0) 0 if the Lyapunov function V(1113954z)

satisfies the following conditions

(a) V(1113954z) is a positive continuous differentiable function(b) ere exist α1 gt 0 α2 gt 0 p isin (0 1) and an open

neighborhood U sub U0 containing the origin suchthat the inequality _V(1113954z)le minus α1V(1113954z) minus αVp(1113954z) holdsthen it can be concluded that the origin is the finitetime stable equilibrium of system (20)

4 Complexity

3 Control Law Design

In this section first we employ a supertwisting observer toestimate the external disturbances in the system Second wepropose a fast nonsingular terminal slidingmode control lawalong with the supertwisting disturbance observer for thespacecraft formation flying system In the last section thenonlinear matrices A and g in (18) and (19) respectively areassumed to be unknown and an adaptive neuroslidingmodel control law for the spacecraft formation flying systemis proposed

31e Fast Nonsingular Terminal SlidingMode Control Lawwith a Supertwisting Disturbance Observer e block dia-gram of the fast nonsingular terminal sliding mode controllaw with the supertwisting disturbance observer is shown inFigure 2 In Figure 2 it can be seen that the overall controllaw consists of two parts a fast nonsingular terminal slidingmode control law along with the supertwisting disturbanceobserver e supertwisting disturbance observer is appliedto estimate the disturbances for the spacecraft formationflying system if the system dynamics are known

311 Supertwisting Disturbance Observer In practical ap-plications the disturbances in the spacecraft formationflying system are fast time-varying disturbances Hence it isdifficult for conventional observers to estimate those dis-turbances In this section a supertwisting disturbance ob-server is employed to estimate those disturbances esupertwisting disturbance observer can estimate the dis-turbance in finite time and also it avoids high-order observergains with fixed bandwidth

According to the attitude-orbit coupling system (15) and(16) the disturbance vector d can be rewritten as follows

d Cminus 12 Mx

e + Ax

e + g minus C1u( 1113857 (22)

Define the state variables as follows

x1 1113946t

0ddt

Cminus 12 Mx

e minus 1113946t

0

ddt

Cminus 12 M1113872 1113873x

e + Cminus 12 Ax

e + g minus C1u( 1113857dt

x2 d

(23)

en the disturbance dynamic system for the spacecraftformation flying is given as

_x1 x2

_x2 _d(24)

For each disturbance di in the spacecraft formationflying system the disturbancersquos model is given as follows

_x1i x2i

_x2i _di i 1 2 6(25)

e supertwisting disturbance observer is given asfollows

_1113957x1i minus λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + 1113957x2i

_1113957x2i minus λ2sign e1i( 1113857(26)

where λ1 and λ2 denote the positive observer gains 1113957x1i and1113957x2i are the estimations of the state vectors x1i and x2i ande1i 1113957x1i minus x1i and e2i 1113957x2i minus x2i are the estimation errorsrespectively

According to the attitude-orbit coupling system (15) and(16) and the supertwisting state observer (26) the estimationerrors can be rewritten as

_e1i minus λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e2i

_e2i minus λ2sign e1i( 1113857 minus d

i(27)

Define the state vectors κ [κ1 κ2]T where

κ1 |e1i|12sign(e1i) κ2 e2i

en the derivatives of the state vectors κ1 and κ2 can bewritten as

_κ1 minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i +12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e2i

_κ2 minus λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i minus d

i

(28)

Similar to [46] the following Lyapunov function for (28)is constructed

Vst κTθ1κ

12

λ21 + 4λ21113872 1113873κ21 minus λ1κ1κ2 + κ22(29)

where θ1 12 (λ21 + 4λ2) minus λ1minus λ1 2

1113890 1113891

en the derivative of Lyapunov function Vst withrespect to the time is given as

_Vst λ21 + 4λ21113872 1113873κ1κ

1 minus λ1κ

1κ2 minus λ1κ1κ

2 + 2κ2κ

2

λ21 + 4λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i +12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e2i1113888 1113889

+λ212

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e22i

minus λ1 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1i minus λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i minus d

i1113874 1113875

minus 2λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minus 2e2id

i

λ21 + 4λ21113872 1113873 minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

+12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i1113888 1113889

+λ212

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e22i

+ λ1λ2 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

+ λ1 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1id

i

minus 2λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minus 2e2id

i

(30)

Furthermore (30) can be written as follows

Complexity 5

V

st minusλ312

+ λ1λ21113888 1113889 e1i

1113868111386811138681113868111386811138681113868111386812

+ λ21e2isign e1i( 1113857 minusλ12

e22i

e1i

1113868111386811138681113868111386811138681113868111386812

+ λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 _di minus 2e2i

_di

minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868 minus 2λ1e2i e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e

22i1113876 1113877

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2 e1i

11138681113868111386811138681113868111386811138681113868sign e1i( 1113857 minus

4λ1

e1i

1113868111386811138681113868111386811138681113868111386812

e2i1113890 1113891 _di

le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868 minus 2λ1e2i e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e

22i1113876 1113877

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2 e1i

11138681113868111386811138681113868111386811138681113868 +

4λ1

e1i

1113868111386811138681113868111386811138681113868111386812

e2i

111386811138681113868111386811138681113868111386811138681113888 1113889 _di

11138681113868111386811138681113868

11138681113868111386811138681113868

(31)

Since κ21 minus 2|κ1||κ2| + κ22 ge 0 (31) can be rewritten as

V

st le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873κ21 minus 2λ1κ1κ2 + κ221113960 1113961

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2κ21 +

4λ1

κ11113868111386811138681113868

1113868111386811138681113868 κ21113868111386811138681113868

11138681113868111386811138681113888 1113889ςi

le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873κ21 minus 2λ1κ1κ2 + κ221113960 1113961

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2κ21 +

2λ1κ21 +

2λ1κ221113888 1113889ςi

(32)

Define the following matrix as

θ2

λ21 + 2λ2 minus2 λ1 + 1( 1113857ςi

λ1minus λ1

minus λ1 1 minus2ςi

λ1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

en (32) can be concluded as

V

st le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812κ

Tθ2κ (34)

From (33) by selecting appropriate observer gains thematrix θ2 can be a positive definite matrixen (34) becomes

V

st le minusλ1λmin θ2( 1113857

2 e1i

1113868111386811138681113868111386811138681113868111386812 κ

2 (35)

where λmin(θ2) is the minimum eigenvalue of θ2θ1 must to be a positive definite matrix to be Lyapunov

matrix hence the observer gains λ1 and λ2 need to be se-lected appropriately so that θ1 gt 0 Let us define the minimaland maximal eigenvalues of θ1 as λmin(θ1)gt 0 andλmax(θ1)gt 0 respectively en we have

λmin θ1( 1113857κ2 leVst le λmax θ1( 1113857κ

2 (36)

Furthermore by observing from the state vectors κ andκ1 the following can be concluded

e1i

1113868111386811138681113868111386811138681113868111386812

κ211113969

le κ (37)

Finally combining (36) with (37) (35) can be derived as

V

st + λ0V12st le 0

λ0 λ1λmin θ1( 1113857

12λmin θ2( 1113857

2λmax θ1( 1113857gt 0

(38)

Based on (37) and Lemma 1 we prove that limt⟶tde1i 0

and limt⟶tde2i 0 with i 1 2 6 Furthermore the

estimations errors of the state vector x1i and x2i can convergein finite time and the convergence time is td 2V12

st λ0

Remark 2 To guarantee that the trajectory errors convergeto zero within finite time the supertwisting disturbanceobserver gains λ1 and λ2 should be selected appropriately sothat θ1 and θ2 are positive definite matrices

312 Fast Nonsingular Terminal Sliding Mode Control Lawwith Supertwisting Disturbance Observer To design a fast

Fast nonsingular terminalsliding mode control law

based on the supertwistingdisturbance observer

Supertwistingdisturbance observer

Spacecraftformation flying

system model

u

d

Desired statesσd ωd ρcd

xe xde

Figure 2 e block diagram of the control law based on the supertwisting disturbance observer

6 Complexity

nonsingular terminal sliding mode control law the fol-lowing sliding mode surface is chosen

s xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 (39)

where isin1 isin2 gt 0 c1 gt nm and 1lt nmlt 2 are all positiveconstants

If the spacecraft formation flying system state errorsreach the sliding mode s 0 then

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 0 (40)

From (15) substituting the equation x

e xde into (40)we have

x

e minus1isin2

1113888 1113889

mn

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 11138571113872 1113873

mn

minus1isin2

1113888 1113889

mn

xmne 1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d1113872 1113873

mn

(41)

In order to analyze the fast convergence of the non-singular fast terminal sliding mode surface assume that theintegration time of the state variable xe denotes as xe(ti) andxe(ti + ts) respectively en integrating the time alongboth sides of (41)

1113946xe ti+ts( )

xe ti( )

1xe

1113888 1113889

mn

dxe minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1 minus 1

1113872 1113873mn

dτ

le minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

dτ

(42)

Furthermore it can be seen from (42) that the trackingerrors of the spacecraft formation flying system can convergeto zero in a limited time ts le (isinmn

2 (n minus m))xe(ti)1minus (mn)

Remark 3 e proposed nonsingular fast terminal slidingmode surface (39) indicates that when the system statevariable xe is far from the equilibrium point the term xe

plays the major role in making the system trajectoryconverge at a fast rate When xe approaches to theequilibrium point the term isin1|xe|

c1d sign(xe) plays the

dominant role to achieve rapid convergence of the statetrajectory

A design procedure for a fast nonsingular terminalsliding mode control law with a supertwisting disturbanceobserver is given in the following theorem

Theorem 1 Consider the spacecraft formation flying systemdescribed by (15) and (16) with

u Cminus 11 Axde + g minus C21113957x2( 1113857 minus C

minus 11 M u0 + u1 + u2( 1113857 (43)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (44)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (45)

u2 α1sign(s) + α2|s|ηsign(s) (46)

and minus α1sign(s) minus α2|s|ηsign(s) is the reaching law with α1 gt 0

α2 gt 0 and ηgt 1 en the proposed control law along withthe supertwisting disturbance observer (26) can guarantee thetracking errors of the spacecraft formation flying systemconverge to zero in finite time

Proof From (39) the derivative of the sliding surface s isgiven as follows

s

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

xde

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

minus Axde minus g(1113872

+ C1u + C2d11138571113857

(47)

Consider the following Lyapunov function

V1(t) 12sTs + 1113944

6

i1V1i (48)

Taking the time derivative of (48) we have

_V1(t) sTs+ 1113944

6

i1

_V1i

sT

xde + isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C211138721113874

middot x2 minus d( 1113857 minus u0 + u1 + u2( 111385711138571113857

minus α0 1113944

6

i1V

121i

(49)

Substituting the proposed control law (44)ndash(46) into(49)

_V1(t) minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857

minus α0 1113944

6

i1V

121i

(50)

Define xdemin min x(nm)minus 1de1 x

(nm)minus 1de61113966 1113967 α1 xdemin

isin2(nm)α1 and α2 xdeminisin2(nm)α2 then

Complexity 7

_V1(t)le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857 minus α0 1113944

6

i1V

121i

le minus α1s minus α2sη+1

+ isin2n

ms xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

x2

minus d minus α0 1113944

6

i1V

121i

(51)

Define αe isin2(nm)|xde|(nm)minus 1d Mminus 1C2x2 minus d and

select αe α1 minus αe ge 0 we have

_V1(t)le minus αes minus α0 1113944

6

i1V

121i (52)

Finally define αm min2

radicαe α01113864 1113865 then (52) can be

rewritten as

_V1(t)le minus αm

12

radic s + 1113944

6

i1V

121i

⎛⎝ ⎞⎠

le minus αm

12s

21113944

6

i1V1i

⎛⎝ ⎞⎠

12

minus αmV1(t)12

(53)

Remark 4 When the system state variable is far from the fastnonsingular terminal sliding mode surface the termα2|s|

ηsign(s) in the reaching law plays the major role On thecontrary the term α1sign(s) in the reaching law dominatesBy choosing different α1 and α2 and selecting appropriateparameters in the fast terminal sliding mode surface theconvergence speed can be controlled Moreover comparedwith the conventional reaching law in [47] the proposedreaching law has faster convergence and better slidingperformance When s 0 and _s 0 the speed of the statevariable approaching the sliding surface is reduced to zerowhich can alleviate the chattering phenomenon

32 Fast Nonsingular Terminal Sliding Mode Control LawBased on the Adaptive Neural Network In this section anadaptive RBF neural network is employed to approximatethe unknown nonlinear A and g in the spacecraft formationflying system (15) and (16) which is not easy to be preciselyestimated in practical situations e block diagram of a fastnonsingular terminal sliding mode control law based on theadaptive neural network is depicted in Figure 3

e structure of RBF neural network with multiple in-puts and multiple outputs is shown in Figure 4

It can be seen that the RBF neural network has threelayers including input layer output layer and hidden layer

Denote x [x1 x2 xn]T as the input and ϕi(x) is theGaussian function of the ith neuron in the hidden layeren the Gaussian function can be expressed as below

ϕi(x) exp minusx minus ci

2

2g2i

⎛⎝ ⎞⎠ i 1 2 6 (54)

where ci is the ith neural radial basis function and gi is thewidth of the ith neural radial basis function In this sectionwe choosefnn to approximate and compensate the unknownnonlinear parts in the spacecraft formation flying systemwhere fnn minus Axde minus g is the unknown nonlinear term Werewrite fnn as follows

fnn WTi ϕi xi( 1113857 + ξi (55)

where Wi denotes the ideal weighting matrix and ξi denotesthe approximation error i 1 2 6

e estimation of fnn can be expressed as follows1113954fnn 1113954W

Ti ϕi xi( 1113857 (56)

where 1113954Wi denotes the estimation of Wi

Theorem 2 Consider the spacecraft formation flying systemdescribed by (15) and (16) with the following control law andthe adaptive updating law of the RBF neural network

u Cminus 11

1113954fnn minus u31113872 1113873 minus Cminus 11 M u0 + u1 + u2( 1113857 (57)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (58)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (59)

u2 α1sign(s) + α2|s|ηsign(s) (60)

u3 M Mminus 1

C1

1sign(s)dm (61)

1113954W

i Πwϕi xi( 1113857ϱi (62)

where ϱi is ith element of the term sTisin2(nm)|xde|(nm)minus 1d Mminus 1

i 1 2 6 and Πw is a positive symmetrical matrix etracking errors of the spacecraft formation flying system willconverge to zero

Proof Define the error of the weighting vector as1113957Wi Wi minus 1113954Wi Consider the following Lyapunov function

V2(t) 12sTs +

12

1113944

n

i1tr 1113957W

T

i Πminus 1w

1113957Wi1113874 1113875 (63)

Taking the derivative of (63) with respect to time andsubstituting the proposed control law (57)ndash(62) and theadaptive updating law (62) then (63) can be converted as

8 Complexity

_V2(t) sTsminus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

sT

minus isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

1sign(s)dm minus Mminus 1

C2d1113872 11138731113874 1113875

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 11113957fnn

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

(64)

Since

sT isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C2

1sign(s)dm + Mminus 1

C2d1113872 11138731113874 1113875

minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873sign si( 1113857dm

+ 11139446

i1siisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873di

le minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113868111386811138681113868

1113868111386811138681113868dm

+ 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2di

11138681113868111386811138681113868111386811138681113868

le 0

(65)

erefore

_V2(t)le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1 1113957fnn + s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WT

i ϕi xi( 1113857 + ξi1113874 1113875

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944n

i1tr 1113957W

T

i ϕi xi( 11138571113874 1113875sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(66)

Since

sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WTi ϕi xi( 11138571113874 1113875

minus 1113944n

i1tr 1113957W

Ti ϕi xi( 11138571113874 1113875s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(67)

(66) can be rewritten as follows

_V2(t)le minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

sTM

minus 11113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

ms

middot xde

11138681113868111386811138681113868111386811138681113868nm(nm)minus 1minus 1d

Mminus 1

1113944

n

i1ξi

(68)

Define ξh isin2(nm)|xde|(nm)minus 1d Mminus 1 1113936

ni1 ξi then we

have_V2(t)le minus ξs (69)

where ξ α1 minus ξherefore it can be concluded that the tracking errors of

the spacecraft formation flying system converge to zero andthe proof is completed

4 Numerical Simulation

In this section the simulation result that shows the effec-tiveness of the supertwisting disturbance observer thenonsingular fast terminal sliding mode control law and theadaptive neurosliding mode control law are presented re-spectively in each section

e parameters in the formation flying system are givenas follows the masses of the target and the chaser spacecraftare mt mc 110 kg the initial orbit parameters of thetarget spacecraft are set as the semimajor axis a 7000 kmthe eccentricity e 001 the orbit inclination i 30deg andthe ascending node is Ω 45deg and apogee and perigee areboth 0deg

e inertial matrix of the chaser spacecraft isJ [3 1 07 1 35 1 05 1 4] kgmiddotm2 and the disturbance tor-que dτ 01[sin(t) minus cos(03t)sin(05t)]T Nm e desiredattitude and position are given as ρcd [2 minus 2 minus 05]Tm andσd 0001[sin(10t) minus cos(10t)sin(10t)]T

41 Supertwisting Disturbance Observer e gains of thesupertwisting disturbance observer are chosen as λ1 2 and

Complexity 9

Fast nonsingular terminalsliding mode control law basedon the adaptive neural network

Spacecraformation flying

system model

u

Desired statesσd ωd ρcd

xe xde

Adaptive updatinglaw

Figure 3 e block diagram of the control law based on the adaptive neural network

x1

x2

xn

h1

h2

hn

y1

y2

yn

Input layer

Hidden layer

Output layer

Figure 4 e structure of RBF neural network

01

0

ndash01

Estim

atio

ner

ror

0 2 4 6 8 10 12 14 16

16

18

18

20

20

Time (s)

001

0

ndash001

(a)

Estim

atio

ner

ror

005

0

ndash005

ndash010 2 4 6 8 10 12 14 16 18 20

Time (s)

16 18 20

01

0

ndash010 05 1

50

ndash5

times10ndash4

(b)

Figure 5 Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbanceobserver

10 Complexity

0 2 4 6 8 10 12 14 16 18 20Time (s)

ndash10

ndash5

0

5

10

15

20

25

x e re

spon

se

16 17 18 19 20ndash1

0

1times10ndash9

Figure 6 e state responses of xe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash20

ndash10

0

10

20

ndash1

0

1times10ndash6

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de re

spon

se

16 17 18 19 20

Figure 7e state responses of xde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash005

0

005

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e resp

onse

Figure 8 e state responses of xe with the nonlinear PD control law (72)

Complexity 11

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

eurort minusμe

r3trt +

dct

mt

(2)

eurorc minusμe

r3crc +

dcc

mc

+fcc

mc

(3)

where mt and mc denote the masses of the target spacecraftand the chaser spacecraft and dct isin R3 and dcc isin R3 denotethe external perturbation forces of the target spacecraft andthe chaser spacecraft respectively fcc isin R3 is the controlforce of the chaser spacecraft rt rt2 and rc rc2 wherert and rc denote the position vectors of the target spacecraftand chaser spacecraft to the geocentric respectively

Define ρ rc minus rt then from (2) and (3) the relativeacceleration vector is obtained as

euroρ minusμe

r3crc +

μe

r3trt +

dcc

mc

minusdct

mt

+fcc

mc

(4)

Furthermore let ρc denote the projection vector in thechaser spacecraft body coordinate frame Fc which denotes

fc minusμe

r3crc +

μe

r3trt

dc dcc minusmcdct

mt

uc fcc

(5)

en

euroρc + ωtimesc ωtimes

c ρc( 1113857 + 2ωtimesc _ρc + _ωtimes

c ρc fc +dc

mc

+uc

mc

(6)

where ωc [ωc1ωc2ωc3]T is the angular velocity of Fc

relative to the inertial coordinate frame FI under Fc frameIn this paper the desired relative position and velocity

under Fc frame are assumed to be ρcd and 0 respectivelyen the error vector can be expressed as ec ρc minus ρcd then

(6) can be written as the EulerndashLagrange equation form toexpress the orbit relative motion

mc euroec + Ac _ec + Bcec + gc uc + dc (7)

where

Ac 2mcωtimesc

Bc mc

minus ω2c2 minus ω2

c3 ωc1ωc2 minus _ωc3 ωc1ωc3 + _ωc2

ωc1ωc2 + _ωc3 minus ω2c1 minus ω2

c3 ωc2ωc3 minus _ωc1

ωc1ωc3 minus _ωc2 ωc2ωc3 + _ωc1 minus ω2c1 minus ω2

c2

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

gc minus mcfc + mc eurordc + Ac _rdc + Bcrdc

(8)

e Modified Rodrigues Parameters (MRPs) are used todescribe the attitude dynamics of the chaser spacecraft whichcan be defined as

σ 1113954q

1 + q ke tan

φ4

(9)

where q [q 1113954qT]T denotes attitude quaternion and ke andφ isin (minus 2π 2π) denote Euler axis and Euler angular of theattitude rotation respectively

en the attitude kinematics and the dynamics of thechaser spacecraft are given as

_σ G(σ)ωc

G(σ) 14

1 minus σTσ1113872 1113873I3times3 + 2σtimes+ 2σσT

1113960 1113961

J _ωc minus ωtimesc Jωc + uτ + dτ

(10)

where J isin R3times3 denotes the inertia matrix of the chaserspacecraft and uτ isin R3 and dτ isin R3 denote the controltorque and the external disturbance torque of the chaserspacecraft respectively

Defining the error attitude σe and the error angularvelocity ωe as

OI

ZI

YI

XI

ycc

xcc

zcc

yct

xct

zct

Earth plane

Solar arrayChaser spacecra

rc

rt

Target spacecra

Solar array

Docking port o ct

Docking port o cc

Figure 1 e relative orbit motion model of the spacecraft formation flying system

Complexity 3

σe 1 minus σTdσd( 1113857σ minus 1 minus σTσ( 1113857σd + 2σtimesσd

1 + σTσσTdσd + 2σTdσ

ωe ωc minus R σe( 1113857ωd

R(σ) I3times3 minus4 1 minus σTσ( 1113857

1 + σTσ( )2 σ

times+

8σtimes

1 + σTσ( )2 σ

times

(11)

where σd isin R3 denotes the desired attitude and ωd isin R3

denotes the desired angular velocity Furthermore the at-titude model of the chaser spacecraft can be expressed as theEulerndashLagrange form

Mτeuroσe + Aτ _σe + gτ Gminus T σe( 1113857dτ + G

minus T σe( 1113857uτ (12)

where

Mτ Gminus T σe( 1113857JG

minus 1 σe( 1113857

G σe( 1113857 14

1 minus σTe σe1113872 1113873I3times3 + 2σtimese + 2σeσ

Te1113960 1113961

Aτ minus Gminus T σe( 1113857 JG

minus 1 σe( 1113857 _σe1113872 1113873timesG

minus 1 σe( 1113857 + J _Gminus 1 σe( 11138571113876 1113877

+ Gminus T σe( 1113857 J R σe( 1113857ωd( 1113857

times+ R σe( 1113857ωd( 1113857

timesJ1113858

minus JR σe( 1113857ωd( 1113857times1113859G

minus 1 σe( 1113857

gτ Gminus T σe( 1113857 R σe( 1113857ωd( 1113857

timesJR σe( 1113857ωd + JR σe( 1113857 _ωd( 1113857

(13)

In order to describe the attitude-orbit coupling dynamicsmotion of the spacecraft formation uniformly define

xe σe

ec

1113890 1113891

u uτ

uc

1113890 1113891

d dτ

dc

1113890 1113891

(14)

en based on (5) and (22) the attitude-orbit couplingmodel is given as

_xe xde (15)

M _xde minus Axde minus g + C1u + C2d (16)

where

M Mτ 03times3

03times3 mcI3times31113890 1113891 (17)

A Aτ 03times3

03times3 Ac

1113890 1113891 (18)

g gτ

Bcec + gc

1113890 1113891 (19)

C1 C2 Gminus T σe( 1113857 03times3

03times3 I3times31113890 1113891 (20)

Remark 1 It can be seen in (12) the attitude and orbit in thespacecraft formation flying system are coupled erefore itis necessary to build the spacecraft coupling motion model(15) and (16) to enhance the control accuracy and efficiencyFrom (15) and (16) it can be seen that the coupling of at-titude and orbit is reflected by the matrix Bc and the vectordτ simultaneously Bc represents the attitude effect on orbitIn addition the gravity gradient torque contains orbit in-formation and the disturbance torque can reflect the in-fluence of the orbit on the attitude thus dτ can reflect theinfluence of the orbit on the attitude

Assumption 1 e external disturbance vectord [d1 d6] in the spacecraft formation flying system(15) and (16) is assumed to satisfy dledm and |di

|le ςiwhere dm gt 0 and ςi gt 0

Assumption 2 All the state variables of the spacecraft for-mation flying system are assumed to be measurable

22 Preliminary Knowledge In this section some relatedlemmas and definitions in this paper are given below

Lemma 1 (see [43]) For _x f(x t) f(0 t) 0 andx isin U0 sub Rn where f U0 times R+⟶ Rn is continuous withrespect to x on an open neighborhood U0 of the origin x 0Suppose that there are a positive-definite function V(x t)

(defined on 1113954U times R+ where 1113954U isin U0 isin Rn is a neighborhood ofthe origin) real numbers cgt 0 and 0lt αlt 1 such that_V(x t) + cVα(x t) is negative semidefinite on 1113954U en

V(x t) is locally finite-time convergent or equivalently andbecomes 0 locally in finite time with its setting timeTleV(x(t0) t0)

1minus αc(1 minus α) for a given initial conditionx(t0) in a neighborhood of the origin in 1113954U

Lemma 2 (see [44]) Suppose Lyapunov function V(1113954z) isdefined on a neighborhood U sub Rn of the origin and _V(1113954z) +

λ0Vℓ le 0 in which λ0 gt 0 0lt ℓ lt 1 en there exists an areaU0 sub Rn such that any V(1113954z) that start from U0 sub Rn canreach V(1113954z) equiv 0 within finite time Tf leV1minus ℓ(0)λ0(1 minus ℓ) inwhich V(0) is the initial time of V(1113954z)

Definition 1 (see [45]) For the autonomous system_1113954z f(1113954z) (21)

where 1113954z isin RN and f(0) 0 if the Lyapunov function V(1113954z)

satisfies the following conditions

(a) V(1113954z) is a positive continuous differentiable function(b) ere exist α1 gt 0 α2 gt 0 p isin (0 1) and an open

neighborhood U sub U0 containing the origin suchthat the inequality _V(1113954z)le minus α1V(1113954z) minus αVp(1113954z) holdsthen it can be concluded that the origin is the finitetime stable equilibrium of system (20)

4 Complexity

3 Control Law Design

In this section first we employ a supertwisting observer toestimate the external disturbances in the system Second wepropose a fast nonsingular terminal slidingmode control lawalong with the supertwisting disturbance observer for thespacecraft formation flying system In the last section thenonlinear matrices A and g in (18) and (19) respectively areassumed to be unknown and an adaptive neuroslidingmodel control law for the spacecraft formation flying systemis proposed

31e Fast Nonsingular Terminal SlidingMode Control Lawwith a Supertwisting Disturbance Observer e block dia-gram of the fast nonsingular terminal sliding mode controllaw with the supertwisting disturbance observer is shown inFigure 2 In Figure 2 it can be seen that the overall controllaw consists of two parts a fast nonsingular terminal slidingmode control law along with the supertwisting disturbanceobserver e supertwisting disturbance observer is appliedto estimate the disturbances for the spacecraft formationflying system if the system dynamics are known

311 Supertwisting Disturbance Observer In practical ap-plications the disturbances in the spacecraft formationflying system are fast time-varying disturbances Hence it isdifficult for conventional observers to estimate those dis-turbances In this section a supertwisting disturbance ob-server is employed to estimate those disturbances esupertwisting disturbance observer can estimate the dis-turbance in finite time and also it avoids high-order observergains with fixed bandwidth

According to the attitude-orbit coupling system (15) and(16) the disturbance vector d can be rewritten as follows

d Cminus 12 Mx

e + Ax

e + g minus C1u( 1113857 (22)

Define the state variables as follows

x1 1113946t

0ddt

Cminus 12 Mx

e minus 1113946t

0

ddt

Cminus 12 M1113872 1113873x

e + Cminus 12 Ax

e + g minus C1u( 1113857dt

x2 d

(23)

en the disturbance dynamic system for the spacecraftformation flying is given as

_x1 x2

_x2 _d(24)

For each disturbance di in the spacecraft formationflying system the disturbancersquos model is given as follows

_x1i x2i

_x2i _di i 1 2 6(25)

e supertwisting disturbance observer is given asfollows

_1113957x1i minus λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + 1113957x2i

_1113957x2i minus λ2sign e1i( 1113857(26)

where λ1 and λ2 denote the positive observer gains 1113957x1i and1113957x2i are the estimations of the state vectors x1i and x2i ande1i 1113957x1i minus x1i and e2i 1113957x2i minus x2i are the estimation errorsrespectively

According to the attitude-orbit coupling system (15) and(16) and the supertwisting state observer (26) the estimationerrors can be rewritten as

_e1i minus λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e2i

_e2i minus λ2sign e1i( 1113857 minus d

i(27)

Define the state vectors κ [κ1 κ2]T where

κ1 |e1i|12sign(e1i) κ2 e2i

en the derivatives of the state vectors κ1 and κ2 can bewritten as

_κ1 minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i +12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e2i

_κ2 minus λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i minus d

i

(28)

Similar to [46] the following Lyapunov function for (28)is constructed

Vst κTθ1κ

12

λ21 + 4λ21113872 1113873κ21 minus λ1κ1κ2 + κ22(29)

where θ1 12 (λ21 + 4λ2) minus λ1minus λ1 2

1113890 1113891

en the derivative of Lyapunov function Vst withrespect to the time is given as

_Vst λ21 + 4λ21113872 1113873κ1κ

1 minus λ1κ

1κ2 minus λ1κ1κ

2 + 2κ2κ

2

λ21 + 4λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i +12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e2i1113888 1113889

+λ212

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e22i

minus λ1 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1i minus λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i minus d

i1113874 1113875

minus 2λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minus 2e2id

i

λ21 + 4λ21113872 1113873 minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

+12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i1113888 1113889

+λ212

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e22i

+ λ1λ2 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

+ λ1 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1id

i

minus 2λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minus 2e2id

i

(30)

Furthermore (30) can be written as follows

Complexity 5

V

st minusλ312

+ λ1λ21113888 1113889 e1i

1113868111386811138681113868111386811138681113868111386812

+ λ21e2isign e1i( 1113857 minusλ12

e22i

e1i

1113868111386811138681113868111386811138681113868111386812

+ λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 _di minus 2e2i

_di

minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868 minus 2λ1e2i e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e

22i1113876 1113877

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2 e1i

11138681113868111386811138681113868111386811138681113868sign e1i( 1113857 minus

4λ1

e1i

1113868111386811138681113868111386811138681113868111386812

e2i1113890 1113891 _di

le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868 minus 2λ1e2i e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e

22i1113876 1113877

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2 e1i

11138681113868111386811138681113868111386811138681113868 +

4λ1

e1i

1113868111386811138681113868111386811138681113868111386812

e2i

111386811138681113868111386811138681113868111386811138681113888 1113889 _di

11138681113868111386811138681113868

11138681113868111386811138681113868

(31)

Since κ21 minus 2|κ1||κ2| + κ22 ge 0 (31) can be rewritten as

V

st le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873κ21 minus 2λ1κ1κ2 + κ221113960 1113961

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2κ21 +

4λ1

κ11113868111386811138681113868

1113868111386811138681113868 κ21113868111386811138681113868

11138681113868111386811138681113888 1113889ςi

le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873κ21 minus 2λ1κ1κ2 + κ221113960 1113961

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2κ21 +

2λ1κ21 +

2λ1κ221113888 1113889ςi

(32)

Define the following matrix as

θ2

λ21 + 2λ2 minus2 λ1 + 1( 1113857ςi

λ1minus λ1

minus λ1 1 minus2ςi

λ1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

en (32) can be concluded as

V

st le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812κ

Tθ2κ (34)

From (33) by selecting appropriate observer gains thematrix θ2 can be a positive definite matrixen (34) becomes

V

st le minusλ1λmin θ2( 1113857

2 e1i

1113868111386811138681113868111386811138681113868111386812 κ

2 (35)

where λmin(θ2) is the minimum eigenvalue of θ2θ1 must to be a positive definite matrix to be Lyapunov

matrix hence the observer gains λ1 and λ2 need to be se-lected appropriately so that θ1 gt 0 Let us define the minimaland maximal eigenvalues of θ1 as λmin(θ1)gt 0 andλmax(θ1)gt 0 respectively en we have

λmin θ1( 1113857κ2 leVst le λmax θ1( 1113857κ

2 (36)

Furthermore by observing from the state vectors κ andκ1 the following can be concluded

e1i

1113868111386811138681113868111386811138681113868111386812

κ211113969

le κ (37)

Finally combining (36) with (37) (35) can be derived as

V

st + λ0V12st le 0

λ0 λ1λmin θ1( 1113857

12λmin θ2( 1113857

2λmax θ1( 1113857gt 0

(38)

Based on (37) and Lemma 1 we prove that limt⟶tde1i 0

and limt⟶tde2i 0 with i 1 2 6 Furthermore the

estimations errors of the state vector x1i and x2i can convergein finite time and the convergence time is td 2V12

st λ0

Remark 2 To guarantee that the trajectory errors convergeto zero within finite time the supertwisting disturbanceobserver gains λ1 and λ2 should be selected appropriately sothat θ1 and θ2 are positive definite matrices

312 Fast Nonsingular Terminal Sliding Mode Control Lawwith Supertwisting Disturbance Observer To design a fast

Fast nonsingular terminalsliding mode control law

based on the supertwistingdisturbance observer

Supertwistingdisturbance observer

Spacecraftformation flying

system model

u

d

Desired statesσd ωd ρcd

xe xde

Figure 2 e block diagram of the control law based on the supertwisting disturbance observer

6 Complexity

nonsingular terminal sliding mode control law the fol-lowing sliding mode surface is chosen

s xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 (39)

where isin1 isin2 gt 0 c1 gt nm and 1lt nmlt 2 are all positiveconstants

If the spacecraft formation flying system state errorsreach the sliding mode s 0 then

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 0 (40)

From (15) substituting the equation x

e xde into (40)we have

x

e minus1isin2

1113888 1113889

mn

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 11138571113872 1113873

mn

minus1isin2

1113888 1113889

mn

xmne 1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d1113872 1113873

mn

(41)

In order to analyze the fast convergence of the non-singular fast terminal sliding mode surface assume that theintegration time of the state variable xe denotes as xe(ti) andxe(ti + ts) respectively en integrating the time alongboth sides of (41)

1113946xe ti+ts( )

xe ti( )

1xe

1113888 1113889

mn

dxe minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1 minus 1

1113872 1113873mn

dτ

le minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

dτ

(42)

Furthermore it can be seen from (42) that the trackingerrors of the spacecraft formation flying system can convergeto zero in a limited time ts le (isinmn

2 (n minus m))xe(ti)1minus (mn)

Remark 3 e proposed nonsingular fast terminal slidingmode surface (39) indicates that when the system statevariable xe is far from the equilibrium point the term xe

plays the major role in making the system trajectoryconverge at a fast rate When xe approaches to theequilibrium point the term isin1|xe|

c1d sign(xe) plays the

dominant role to achieve rapid convergence of the statetrajectory

A design procedure for a fast nonsingular terminalsliding mode control law with a supertwisting disturbanceobserver is given in the following theorem

Theorem 1 Consider the spacecraft formation flying systemdescribed by (15) and (16) with

u Cminus 11 Axde + g minus C21113957x2( 1113857 minus C

minus 11 M u0 + u1 + u2( 1113857 (43)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (44)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (45)

u2 α1sign(s) + α2|s|ηsign(s) (46)

and minus α1sign(s) minus α2|s|ηsign(s) is the reaching law with α1 gt 0

α2 gt 0 and ηgt 1 en the proposed control law along withthe supertwisting disturbance observer (26) can guarantee thetracking errors of the spacecraft formation flying systemconverge to zero in finite time

Proof From (39) the derivative of the sliding surface s isgiven as follows

s

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

xde

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

minus Axde minus g(1113872

+ C1u + C2d11138571113857

(47)

Consider the following Lyapunov function

V1(t) 12sTs + 1113944

6

i1V1i (48)

Taking the time derivative of (48) we have

_V1(t) sTs+ 1113944

6

i1

_V1i

sT

xde + isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C211138721113874

middot x2 minus d( 1113857 minus u0 + u1 + u2( 111385711138571113857

minus α0 1113944

6

i1V

121i

(49)

Substituting the proposed control law (44)ndash(46) into(49)

_V1(t) minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857

minus α0 1113944

6

i1V

121i

(50)

Define xdemin min x(nm)minus 1de1 x

(nm)minus 1de61113966 1113967 α1 xdemin

isin2(nm)α1 and α2 xdeminisin2(nm)α2 then

Complexity 7

_V1(t)le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857 minus α0 1113944

6

i1V

121i

le minus α1s minus α2sη+1

+ isin2n

ms xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

x2

minus d minus α0 1113944

6

i1V

121i

(51)

Define αe isin2(nm)|xde|(nm)minus 1d Mminus 1C2x2 minus d and

select αe α1 minus αe ge 0 we have

_V1(t)le minus αes minus α0 1113944

6

i1V

121i (52)

Finally define αm min2

radicαe α01113864 1113865 then (52) can be

rewritten as

_V1(t)le minus αm

12

radic s + 1113944

6

i1V

121i

⎛⎝ ⎞⎠

le minus αm

12s

21113944

6

i1V1i

⎛⎝ ⎞⎠

12

minus αmV1(t)12

(53)

Remark 4 When the system state variable is far from the fastnonsingular terminal sliding mode surface the termα2|s|

ηsign(s) in the reaching law plays the major role On thecontrary the term α1sign(s) in the reaching law dominatesBy choosing different α1 and α2 and selecting appropriateparameters in the fast terminal sliding mode surface theconvergence speed can be controlled Moreover comparedwith the conventional reaching law in [47] the proposedreaching law has faster convergence and better slidingperformance When s 0 and _s 0 the speed of the statevariable approaching the sliding surface is reduced to zerowhich can alleviate the chattering phenomenon

32 Fast Nonsingular Terminal Sliding Mode Control LawBased on the Adaptive Neural Network In this section anadaptive RBF neural network is employed to approximatethe unknown nonlinear A and g in the spacecraft formationflying system (15) and (16) which is not easy to be preciselyestimated in practical situations e block diagram of a fastnonsingular terminal sliding mode control law based on theadaptive neural network is depicted in Figure 3

e structure of RBF neural network with multiple in-puts and multiple outputs is shown in Figure 4

It can be seen that the RBF neural network has threelayers including input layer output layer and hidden layer

Denote x [x1 x2 xn]T as the input and ϕi(x) is theGaussian function of the ith neuron in the hidden layeren the Gaussian function can be expressed as below

ϕi(x) exp minusx minus ci

2

2g2i

⎛⎝ ⎞⎠ i 1 2 6 (54)

where ci is the ith neural radial basis function and gi is thewidth of the ith neural radial basis function In this sectionwe choosefnn to approximate and compensate the unknownnonlinear parts in the spacecraft formation flying systemwhere fnn minus Axde minus g is the unknown nonlinear term Werewrite fnn as follows

fnn WTi ϕi xi( 1113857 + ξi (55)

where Wi denotes the ideal weighting matrix and ξi denotesthe approximation error i 1 2 6

e estimation of fnn can be expressed as follows1113954fnn 1113954W

Ti ϕi xi( 1113857 (56)

where 1113954Wi denotes the estimation of Wi

Theorem 2 Consider the spacecraft formation flying systemdescribed by (15) and (16) with the following control law andthe adaptive updating law of the RBF neural network

u Cminus 11

1113954fnn minus u31113872 1113873 minus Cminus 11 M u0 + u1 + u2( 1113857 (57)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (58)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (59)

u2 α1sign(s) + α2|s|ηsign(s) (60)

u3 M Mminus 1

C1

1sign(s)dm (61)

1113954W

i Πwϕi xi( 1113857ϱi (62)

where ϱi is ith element of the term sTisin2(nm)|xde|(nm)minus 1d Mminus 1

i 1 2 6 and Πw is a positive symmetrical matrix etracking errors of the spacecraft formation flying system willconverge to zero

Proof Define the error of the weighting vector as1113957Wi Wi minus 1113954Wi Consider the following Lyapunov function

V2(t) 12sTs +

12

1113944

n

i1tr 1113957W

T

i Πminus 1w

1113957Wi1113874 1113875 (63)

Taking the derivative of (63) with respect to time andsubstituting the proposed control law (57)ndash(62) and theadaptive updating law (62) then (63) can be converted as

8 Complexity

_V2(t) sTsminus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

sT

minus isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

1sign(s)dm minus Mminus 1

C2d1113872 11138731113874 1113875

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 11113957fnn

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

(64)

Since

sT isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C2

1sign(s)dm + Mminus 1

C2d1113872 11138731113874 1113875

minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873sign si( 1113857dm

+ 11139446

i1siisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873di

le minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113868111386811138681113868

1113868111386811138681113868dm

+ 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2di

11138681113868111386811138681113868111386811138681113868

le 0

(65)

erefore

_V2(t)le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1 1113957fnn + s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WT

i ϕi xi( 1113857 + ξi1113874 1113875

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944n

i1tr 1113957W

T

i ϕi xi( 11138571113874 1113875sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(66)

Since

sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WTi ϕi xi( 11138571113874 1113875

minus 1113944n

i1tr 1113957W

Ti ϕi xi( 11138571113874 1113875s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(67)

(66) can be rewritten as follows

_V2(t)le minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

sTM

minus 11113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

ms

middot xde

11138681113868111386811138681113868111386811138681113868nm(nm)minus 1minus 1d

Mminus 1

1113944

n

i1ξi

(68)

Define ξh isin2(nm)|xde|(nm)minus 1d Mminus 1 1113936

ni1 ξi then we

have_V2(t)le minus ξs (69)

where ξ α1 minus ξherefore it can be concluded that the tracking errors of

the spacecraft formation flying system converge to zero andthe proof is completed

4 Numerical Simulation

In this section the simulation result that shows the effec-tiveness of the supertwisting disturbance observer thenonsingular fast terminal sliding mode control law and theadaptive neurosliding mode control law are presented re-spectively in each section

e parameters in the formation flying system are givenas follows the masses of the target and the chaser spacecraftare mt mc 110 kg the initial orbit parameters of thetarget spacecraft are set as the semimajor axis a 7000 kmthe eccentricity e 001 the orbit inclination i 30deg andthe ascending node is Ω 45deg and apogee and perigee areboth 0deg

e inertial matrix of the chaser spacecraft isJ [3 1 07 1 35 1 05 1 4] kgmiddotm2 and the disturbance tor-que dτ 01[sin(t) minus cos(03t)sin(05t)]T Nm e desiredattitude and position are given as ρcd [2 minus 2 minus 05]Tm andσd 0001[sin(10t) minus cos(10t)sin(10t)]T

41 Supertwisting Disturbance Observer e gains of thesupertwisting disturbance observer are chosen as λ1 2 and

Complexity 9

Fast nonsingular terminalsliding mode control law basedon the adaptive neural network

Spacecraformation flying

system model

u

Desired statesσd ωd ρcd

xe xde

Adaptive updatinglaw

Figure 3 e block diagram of the control law based on the adaptive neural network

x1

x2

xn

h1

h2

hn

y1

y2

yn

Input layer

Hidden layer

Output layer

Figure 4 e structure of RBF neural network

01

0

ndash01

Estim

atio

ner

ror

0 2 4 6 8 10 12 14 16

16

18

18

20

20

Time (s)

001

0

ndash001

(a)

Estim

atio

ner

ror

005

0

ndash005

ndash010 2 4 6 8 10 12 14 16 18 20

Time (s)

16 18 20

01

0

ndash010 05 1

50

ndash5

times10ndash4

(b)

Figure 5 Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbanceobserver

10 Complexity

0 2 4 6 8 10 12 14 16 18 20Time (s)

ndash10

ndash5

0

5

10

15

20

25

x e re

spon

se

16 17 18 19 20ndash1

0

1times10ndash9

Figure 6 e state responses of xe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash20

ndash10

0

10

20

ndash1

0

1times10ndash6

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de re

spon

se

16 17 18 19 20

Figure 7e state responses of xde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash005

0

005

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e resp

onse

Figure 8 e state responses of xe with the nonlinear PD control law (72)

Complexity 11

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

σe 1 minus σTdσd( 1113857σ minus 1 minus σTσ( 1113857σd + 2σtimesσd

1 + σTσσTdσd + 2σTdσ

ωe ωc minus R σe( 1113857ωd

R(σ) I3times3 minus4 1 minus σTσ( 1113857

1 + σTσ( )2 σ

times+

8σtimes

1 + σTσ( )2 σ

times

(11)

where σd isin R3 denotes the desired attitude and ωd isin R3

denotes the desired angular velocity Furthermore the at-titude model of the chaser spacecraft can be expressed as theEulerndashLagrange form

Mτeuroσe + Aτ _σe + gτ Gminus T σe( 1113857dτ + G

minus T σe( 1113857uτ (12)

where

Mτ Gminus T σe( 1113857JG

minus 1 σe( 1113857

G σe( 1113857 14

1 minus σTe σe1113872 1113873I3times3 + 2σtimese + 2σeσ

Te1113960 1113961

Aτ minus Gminus T σe( 1113857 JG

minus 1 σe( 1113857 _σe1113872 1113873timesG

minus 1 σe( 1113857 + J _Gminus 1 σe( 11138571113876 1113877

+ Gminus T σe( 1113857 J R σe( 1113857ωd( 1113857

times+ R σe( 1113857ωd( 1113857

timesJ1113858

minus JR σe( 1113857ωd( 1113857times1113859G

minus 1 σe( 1113857

gτ Gminus T σe( 1113857 R σe( 1113857ωd( 1113857

timesJR σe( 1113857ωd + JR σe( 1113857 _ωd( 1113857

(13)

In order to describe the attitude-orbit coupling dynamicsmotion of the spacecraft formation uniformly define

xe σe

ec

1113890 1113891

u uτ

uc

1113890 1113891

d dτ

dc

1113890 1113891

(14)

en based on (5) and (22) the attitude-orbit couplingmodel is given as

_xe xde (15)

M _xde minus Axde minus g + C1u + C2d (16)

where

M Mτ 03times3

03times3 mcI3times31113890 1113891 (17)

A Aτ 03times3

03times3 Ac

1113890 1113891 (18)

g gτ

Bcec + gc

1113890 1113891 (19)

C1 C2 Gminus T σe( 1113857 03times3

03times3 I3times31113890 1113891 (20)

Remark 1 It can be seen in (12) the attitude and orbit in thespacecraft formation flying system are coupled erefore itis necessary to build the spacecraft coupling motion model(15) and (16) to enhance the control accuracy and efficiencyFrom (15) and (16) it can be seen that the coupling of at-titude and orbit is reflected by the matrix Bc and the vectordτ simultaneously Bc represents the attitude effect on orbitIn addition the gravity gradient torque contains orbit in-formation and the disturbance torque can reflect the in-fluence of the orbit on the attitude thus dτ can reflect theinfluence of the orbit on the attitude

Assumption 1 e external disturbance vectord [d1 d6] in the spacecraft formation flying system(15) and (16) is assumed to satisfy dledm and |di

|le ςiwhere dm gt 0 and ςi gt 0

Assumption 2 All the state variables of the spacecraft for-mation flying system are assumed to be measurable

22 Preliminary Knowledge In this section some relatedlemmas and definitions in this paper are given below

Lemma 1 (see [43]) For _x f(x t) f(0 t) 0 andx isin U0 sub Rn where f U0 times R+⟶ Rn is continuous withrespect to x on an open neighborhood U0 of the origin x 0Suppose that there are a positive-definite function V(x t)

(defined on 1113954U times R+ where 1113954U isin U0 isin Rn is a neighborhood ofthe origin) real numbers cgt 0 and 0lt αlt 1 such that_V(x t) + cVα(x t) is negative semidefinite on 1113954U en

V(x t) is locally finite-time convergent or equivalently andbecomes 0 locally in finite time with its setting timeTleV(x(t0) t0)

1minus αc(1 minus α) for a given initial conditionx(t0) in a neighborhood of the origin in 1113954U

Lemma 2 (see [44]) Suppose Lyapunov function V(1113954z) isdefined on a neighborhood U sub Rn of the origin and _V(1113954z) +

λ0Vℓ le 0 in which λ0 gt 0 0lt ℓ lt 1 en there exists an areaU0 sub Rn such that any V(1113954z) that start from U0 sub Rn canreach V(1113954z) equiv 0 within finite time Tf leV1minus ℓ(0)λ0(1 minus ℓ) inwhich V(0) is the initial time of V(1113954z)

Definition 1 (see [45]) For the autonomous system_1113954z f(1113954z) (21)

where 1113954z isin RN and f(0) 0 if the Lyapunov function V(1113954z)

satisfies the following conditions

(a) V(1113954z) is a positive continuous differentiable function(b) ere exist α1 gt 0 α2 gt 0 p isin (0 1) and an open

neighborhood U sub U0 containing the origin suchthat the inequality _V(1113954z)le minus α1V(1113954z) minus αVp(1113954z) holdsthen it can be concluded that the origin is the finitetime stable equilibrium of system (20)

4 Complexity

3 Control Law Design

In this section first we employ a supertwisting observer toestimate the external disturbances in the system Second wepropose a fast nonsingular terminal slidingmode control lawalong with the supertwisting disturbance observer for thespacecraft formation flying system In the last section thenonlinear matrices A and g in (18) and (19) respectively areassumed to be unknown and an adaptive neuroslidingmodel control law for the spacecraft formation flying systemis proposed

31e Fast Nonsingular Terminal SlidingMode Control Lawwith a Supertwisting Disturbance Observer e block dia-gram of the fast nonsingular terminal sliding mode controllaw with the supertwisting disturbance observer is shown inFigure 2 In Figure 2 it can be seen that the overall controllaw consists of two parts a fast nonsingular terminal slidingmode control law along with the supertwisting disturbanceobserver e supertwisting disturbance observer is appliedto estimate the disturbances for the spacecraft formationflying system if the system dynamics are known

311 Supertwisting Disturbance Observer In practical ap-plications the disturbances in the spacecraft formationflying system are fast time-varying disturbances Hence it isdifficult for conventional observers to estimate those dis-turbances In this section a supertwisting disturbance ob-server is employed to estimate those disturbances esupertwisting disturbance observer can estimate the dis-turbance in finite time and also it avoids high-order observergains with fixed bandwidth

According to the attitude-orbit coupling system (15) and(16) the disturbance vector d can be rewritten as follows

d Cminus 12 Mx

e + Ax

e + g minus C1u( 1113857 (22)

Define the state variables as follows

x1 1113946t

0ddt

Cminus 12 Mx

e minus 1113946t

0

ddt

Cminus 12 M1113872 1113873x

e + Cminus 12 Ax

e + g minus C1u( 1113857dt

x2 d

(23)

en the disturbance dynamic system for the spacecraftformation flying is given as

_x1 x2

_x2 _d(24)

For each disturbance di in the spacecraft formationflying system the disturbancersquos model is given as follows

_x1i x2i

_x2i _di i 1 2 6(25)

e supertwisting disturbance observer is given asfollows

_1113957x1i minus λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + 1113957x2i

_1113957x2i minus λ2sign e1i( 1113857(26)

where λ1 and λ2 denote the positive observer gains 1113957x1i and1113957x2i are the estimations of the state vectors x1i and x2i ande1i 1113957x1i minus x1i and e2i 1113957x2i minus x2i are the estimation errorsrespectively

According to the attitude-orbit coupling system (15) and(16) and the supertwisting state observer (26) the estimationerrors can be rewritten as

_e1i minus λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e2i

_e2i minus λ2sign e1i( 1113857 minus d

i(27)

Define the state vectors κ [κ1 κ2]T where

κ1 |e1i|12sign(e1i) κ2 e2i

en the derivatives of the state vectors κ1 and κ2 can bewritten as

_κ1 minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i +12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e2i

_κ2 minus λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i minus d

i

(28)

Similar to [46] the following Lyapunov function for (28)is constructed

Vst κTθ1κ

12

λ21 + 4λ21113872 1113873κ21 minus λ1κ1κ2 + κ22(29)

where θ1 12 (λ21 + 4λ2) minus λ1minus λ1 2

1113890 1113891

en the derivative of Lyapunov function Vst withrespect to the time is given as

_Vst λ21 + 4λ21113872 1113873κ1κ

1 minus λ1κ

1κ2 minus λ1κ1κ

2 + 2κ2κ

2

λ21 + 4λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i +12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e2i1113888 1113889

+λ212

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e22i

minus λ1 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1i minus λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i minus d

i1113874 1113875

minus 2λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minus 2e2id

i

λ21 + 4λ21113872 1113873 minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

+12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i1113888 1113889

+λ212

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e22i

+ λ1λ2 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

+ λ1 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1id

i

minus 2λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minus 2e2id

i

(30)

Furthermore (30) can be written as follows

Complexity 5

V

st minusλ312

+ λ1λ21113888 1113889 e1i

1113868111386811138681113868111386811138681113868111386812

+ λ21e2isign e1i( 1113857 minusλ12

e22i

e1i

1113868111386811138681113868111386811138681113868111386812

+ λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 _di minus 2e2i

_di

minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868 minus 2λ1e2i e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e

22i1113876 1113877

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2 e1i

11138681113868111386811138681113868111386811138681113868sign e1i( 1113857 minus

4λ1

e1i

1113868111386811138681113868111386811138681113868111386812

e2i1113890 1113891 _di

le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868 minus 2λ1e2i e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e

22i1113876 1113877

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2 e1i

11138681113868111386811138681113868111386811138681113868 +

4λ1

e1i

1113868111386811138681113868111386811138681113868111386812

e2i

111386811138681113868111386811138681113868111386811138681113888 1113889 _di

11138681113868111386811138681113868

11138681113868111386811138681113868

(31)

Since κ21 minus 2|κ1||κ2| + κ22 ge 0 (31) can be rewritten as

V

st le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873κ21 minus 2λ1κ1κ2 + κ221113960 1113961

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2κ21 +

4λ1

κ11113868111386811138681113868

1113868111386811138681113868 κ21113868111386811138681113868

11138681113868111386811138681113888 1113889ςi

le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873κ21 minus 2λ1κ1κ2 + κ221113960 1113961

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2κ21 +

2λ1κ21 +

2λ1κ221113888 1113889ςi

(32)

Define the following matrix as

θ2

λ21 + 2λ2 minus2 λ1 + 1( 1113857ςi

λ1minus λ1

minus λ1 1 minus2ςi

λ1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

en (32) can be concluded as

V

st le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812κ

Tθ2κ (34)

From (33) by selecting appropriate observer gains thematrix θ2 can be a positive definite matrixen (34) becomes

V

st le minusλ1λmin θ2( 1113857

2 e1i

1113868111386811138681113868111386811138681113868111386812 κ

2 (35)

where λmin(θ2) is the minimum eigenvalue of θ2θ1 must to be a positive definite matrix to be Lyapunov

matrix hence the observer gains λ1 and λ2 need to be se-lected appropriately so that θ1 gt 0 Let us define the minimaland maximal eigenvalues of θ1 as λmin(θ1)gt 0 andλmax(θ1)gt 0 respectively en we have

λmin θ1( 1113857κ2 leVst le λmax θ1( 1113857κ

2 (36)

Furthermore by observing from the state vectors κ andκ1 the following can be concluded

e1i

1113868111386811138681113868111386811138681113868111386812

κ211113969

le κ (37)

Finally combining (36) with (37) (35) can be derived as

V

st + λ0V12st le 0

λ0 λ1λmin θ1( 1113857

12λmin θ2( 1113857

2λmax θ1( 1113857gt 0

(38)

Based on (37) and Lemma 1 we prove that limt⟶tde1i 0

and limt⟶tde2i 0 with i 1 2 6 Furthermore the

estimations errors of the state vector x1i and x2i can convergein finite time and the convergence time is td 2V12

st λ0

Remark 2 To guarantee that the trajectory errors convergeto zero within finite time the supertwisting disturbanceobserver gains λ1 and λ2 should be selected appropriately sothat θ1 and θ2 are positive definite matrices

312 Fast Nonsingular Terminal Sliding Mode Control Lawwith Supertwisting Disturbance Observer To design a fast

Fast nonsingular terminalsliding mode control law

based on the supertwistingdisturbance observer

Supertwistingdisturbance observer

Spacecraftformation flying

system model

u

d

Desired statesσd ωd ρcd

xe xde

Figure 2 e block diagram of the control law based on the supertwisting disturbance observer

6 Complexity

nonsingular terminal sliding mode control law the fol-lowing sliding mode surface is chosen

s xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 (39)

where isin1 isin2 gt 0 c1 gt nm and 1lt nmlt 2 are all positiveconstants

If the spacecraft formation flying system state errorsreach the sliding mode s 0 then

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 0 (40)

From (15) substituting the equation x

e xde into (40)we have

x

e minus1isin2

1113888 1113889

mn

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 11138571113872 1113873

mn

minus1isin2

1113888 1113889

mn

xmne 1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d1113872 1113873

mn

(41)

In order to analyze the fast convergence of the non-singular fast terminal sliding mode surface assume that theintegration time of the state variable xe denotes as xe(ti) andxe(ti + ts) respectively en integrating the time alongboth sides of (41)

1113946xe ti+ts( )

xe ti( )

1xe

1113888 1113889

mn

dxe minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1 minus 1

1113872 1113873mn

dτ

le minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

dτ

(42)

Furthermore it can be seen from (42) that the trackingerrors of the spacecraft formation flying system can convergeto zero in a limited time ts le (isinmn

2 (n minus m))xe(ti)1minus (mn)

Remark 3 e proposed nonsingular fast terminal slidingmode surface (39) indicates that when the system statevariable xe is far from the equilibrium point the term xe

plays the major role in making the system trajectoryconverge at a fast rate When xe approaches to theequilibrium point the term isin1|xe|

c1d sign(xe) plays the

dominant role to achieve rapid convergence of the statetrajectory

A design procedure for a fast nonsingular terminalsliding mode control law with a supertwisting disturbanceobserver is given in the following theorem

Theorem 1 Consider the spacecraft formation flying systemdescribed by (15) and (16) with

u Cminus 11 Axde + g minus C21113957x2( 1113857 minus C

minus 11 M u0 + u1 + u2( 1113857 (43)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (44)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (45)

u2 α1sign(s) + α2|s|ηsign(s) (46)

and minus α1sign(s) minus α2|s|ηsign(s) is the reaching law with α1 gt 0

α2 gt 0 and ηgt 1 en the proposed control law along withthe supertwisting disturbance observer (26) can guarantee thetracking errors of the spacecraft formation flying systemconverge to zero in finite time

Proof From (39) the derivative of the sliding surface s isgiven as follows

s

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

xde

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

minus Axde minus g(1113872

+ C1u + C2d11138571113857

(47)

Consider the following Lyapunov function

V1(t) 12sTs + 1113944

6

i1V1i (48)

Taking the time derivative of (48) we have

_V1(t) sTs+ 1113944

6

i1

_V1i

sT

xde + isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C211138721113874

middot x2 minus d( 1113857 minus u0 + u1 + u2( 111385711138571113857

minus α0 1113944

6

i1V

121i

(49)

Substituting the proposed control law (44)ndash(46) into(49)

_V1(t) minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857

minus α0 1113944

6

i1V

121i

(50)

Define xdemin min x(nm)minus 1de1 x

(nm)minus 1de61113966 1113967 α1 xdemin

isin2(nm)α1 and α2 xdeminisin2(nm)α2 then

Complexity 7

_V1(t)le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857 minus α0 1113944

6

i1V

121i

le minus α1s minus α2sη+1

+ isin2n

ms xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

x2

minus d minus α0 1113944

6

i1V

121i

(51)

Define αe isin2(nm)|xde|(nm)minus 1d Mminus 1C2x2 minus d and

select αe α1 minus αe ge 0 we have

_V1(t)le minus αes minus α0 1113944

6

i1V

121i (52)

Finally define αm min2

radicαe α01113864 1113865 then (52) can be

rewritten as

_V1(t)le minus αm

12

radic s + 1113944

6

i1V

121i

⎛⎝ ⎞⎠

le minus αm

12s

21113944

6

i1V1i

⎛⎝ ⎞⎠

12

minus αmV1(t)12

(53)

Remark 4 When the system state variable is far from the fastnonsingular terminal sliding mode surface the termα2|s|

ηsign(s) in the reaching law plays the major role On thecontrary the term α1sign(s) in the reaching law dominatesBy choosing different α1 and α2 and selecting appropriateparameters in the fast terminal sliding mode surface theconvergence speed can be controlled Moreover comparedwith the conventional reaching law in [47] the proposedreaching law has faster convergence and better slidingperformance When s 0 and _s 0 the speed of the statevariable approaching the sliding surface is reduced to zerowhich can alleviate the chattering phenomenon

32 Fast Nonsingular Terminal Sliding Mode Control LawBased on the Adaptive Neural Network In this section anadaptive RBF neural network is employed to approximatethe unknown nonlinear A and g in the spacecraft formationflying system (15) and (16) which is not easy to be preciselyestimated in practical situations e block diagram of a fastnonsingular terminal sliding mode control law based on theadaptive neural network is depicted in Figure 3

e structure of RBF neural network with multiple in-puts and multiple outputs is shown in Figure 4

It can be seen that the RBF neural network has threelayers including input layer output layer and hidden layer

Denote x [x1 x2 xn]T as the input and ϕi(x) is theGaussian function of the ith neuron in the hidden layeren the Gaussian function can be expressed as below

ϕi(x) exp minusx minus ci

2

2g2i

⎛⎝ ⎞⎠ i 1 2 6 (54)

where ci is the ith neural radial basis function and gi is thewidth of the ith neural radial basis function In this sectionwe choosefnn to approximate and compensate the unknownnonlinear parts in the spacecraft formation flying systemwhere fnn minus Axde minus g is the unknown nonlinear term Werewrite fnn as follows

fnn WTi ϕi xi( 1113857 + ξi (55)

where Wi denotes the ideal weighting matrix and ξi denotesthe approximation error i 1 2 6

e estimation of fnn can be expressed as follows1113954fnn 1113954W

Ti ϕi xi( 1113857 (56)

where 1113954Wi denotes the estimation of Wi

Theorem 2 Consider the spacecraft formation flying systemdescribed by (15) and (16) with the following control law andthe adaptive updating law of the RBF neural network

u Cminus 11

1113954fnn minus u31113872 1113873 minus Cminus 11 M u0 + u1 + u2( 1113857 (57)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (58)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (59)

u2 α1sign(s) + α2|s|ηsign(s) (60)

u3 M Mminus 1

C1

1sign(s)dm (61)

1113954W

i Πwϕi xi( 1113857ϱi (62)

where ϱi is ith element of the term sTisin2(nm)|xde|(nm)minus 1d Mminus 1

i 1 2 6 and Πw is a positive symmetrical matrix etracking errors of the spacecraft formation flying system willconverge to zero

Proof Define the error of the weighting vector as1113957Wi Wi minus 1113954Wi Consider the following Lyapunov function

V2(t) 12sTs +

12

1113944

n

i1tr 1113957W

T

i Πminus 1w

1113957Wi1113874 1113875 (63)

Taking the derivative of (63) with respect to time andsubstituting the proposed control law (57)ndash(62) and theadaptive updating law (62) then (63) can be converted as

8 Complexity

_V2(t) sTsminus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

sT

minus isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

1sign(s)dm minus Mminus 1

C2d1113872 11138731113874 1113875

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 11113957fnn

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

(64)

Since

sT isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C2

1sign(s)dm + Mminus 1

C2d1113872 11138731113874 1113875

minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873sign si( 1113857dm

+ 11139446

i1siisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873di

le minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113868111386811138681113868

1113868111386811138681113868dm

+ 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2di

11138681113868111386811138681113868111386811138681113868

le 0

(65)

erefore

_V2(t)le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1 1113957fnn + s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WT

i ϕi xi( 1113857 + ξi1113874 1113875

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944n

i1tr 1113957W

T

i ϕi xi( 11138571113874 1113875sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(66)

Since

sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WTi ϕi xi( 11138571113874 1113875

minus 1113944n

i1tr 1113957W

Ti ϕi xi( 11138571113874 1113875s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(67)

(66) can be rewritten as follows

_V2(t)le minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

sTM

minus 11113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

ms

middot xde

11138681113868111386811138681113868111386811138681113868nm(nm)minus 1minus 1d

Mminus 1

1113944

n

i1ξi

(68)

Define ξh isin2(nm)|xde|(nm)minus 1d Mminus 1 1113936

ni1 ξi then we

have_V2(t)le minus ξs (69)

where ξ α1 minus ξherefore it can be concluded that the tracking errors of

the spacecraft formation flying system converge to zero andthe proof is completed

4 Numerical Simulation

In this section the simulation result that shows the effec-tiveness of the supertwisting disturbance observer thenonsingular fast terminal sliding mode control law and theadaptive neurosliding mode control law are presented re-spectively in each section

e parameters in the formation flying system are givenas follows the masses of the target and the chaser spacecraftare mt mc 110 kg the initial orbit parameters of thetarget spacecraft are set as the semimajor axis a 7000 kmthe eccentricity e 001 the orbit inclination i 30deg andthe ascending node is Ω 45deg and apogee and perigee areboth 0deg

e inertial matrix of the chaser spacecraft isJ [3 1 07 1 35 1 05 1 4] kgmiddotm2 and the disturbance tor-que dτ 01[sin(t) minus cos(03t)sin(05t)]T Nm e desiredattitude and position are given as ρcd [2 minus 2 minus 05]Tm andσd 0001[sin(10t) minus cos(10t)sin(10t)]T

41 Supertwisting Disturbance Observer e gains of thesupertwisting disturbance observer are chosen as λ1 2 and

Complexity 9

Fast nonsingular terminalsliding mode control law basedon the adaptive neural network

Spacecraformation flying

system model

u

Desired statesσd ωd ρcd

xe xde

Adaptive updatinglaw

Figure 3 e block diagram of the control law based on the adaptive neural network

x1

x2

xn

h1

h2

hn

y1

y2

yn

Input layer

Hidden layer

Output layer

Figure 4 e structure of RBF neural network

01

0

ndash01

Estim

atio

ner

ror

0 2 4 6 8 10 12 14 16

16

18

18

20

20

Time (s)

001

0

ndash001

(a)

Estim

atio

ner

ror

005

0

ndash005

ndash010 2 4 6 8 10 12 14 16 18 20

Time (s)

16 18 20

01

0

ndash010 05 1

50

ndash5

times10ndash4

(b)

Figure 5 Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbanceobserver

10 Complexity

0 2 4 6 8 10 12 14 16 18 20Time (s)

ndash10

ndash5

0

5

10

15

20

25

x e re

spon

se

16 17 18 19 20ndash1

0

1times10ndash9

Figure 6 e state responses of xe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash20

ndash10

0

10

20

ndash1

0

1times10ndash6

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de re

spon

se

16 17 18 19 20

Figure 7e state responses of xde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash005

0

005

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e resp

onse

Figure 8 e state responses of xe with the nonlinear PD control law (72)

Complexity 11

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

3 Control Law Design

In this section first we employ a supertwisting observer toestimate the external disturbances in the system Second wepropose a fast nonsingular terminal slidingmode control lawalong with the supertwisting disturbance observer for thespacecraft formation flying system In the last section thenonlinear matrices A and g in (18) and (19) respectively areassumed to be unknown and an adaptive neuroslidingmodel control law for the spacecraft formation flying systemis proposed

31e Fast Nonsingular Terminal SlidingMode Control Lawwith a Supertwisting Disturbance Observer e block dia-gram of the fast nonsingular terminal sliding mode controllaw with the supertwisting disturbance observer is shown inFigure 2 In Figure 2 it can be seen that the overall controllaw consists of two parts a fast nonsingular terminal slidingmode control law along with the supertwisting disturbanceobserver e supertwisting disturbance observer is appliedto estimate the disturbances for the spacecraft formationflying system if the system dynamics are known

311 Supertwisting Disturbance Observer In practical ap-plications the disturbances in the spacecraft formationflying system are fast time-varying disturbances Hence it isdifficult for conventional observers to estimate those dis-turbances In this section a supertwisting disturbance ob-server is employed to estimate those disturbances esupertwisting disturbance observer can estimate the dis-turbance in finite time and also it avoids high-order observergains with fixed bandwidth

According to the attitude-orbit coupling system (15) and(16) the disturbance vector d can be rewritten as follows

d Cminus 12 Mx

e + Ax

e + g minus C1u( 1113857 (22)

Define the state variables as follows

x1 1113946t

0ddt

Cminus 12 Mx

e minus 1113946t

0

ddt

Cminus 12 M1113872 1113873x

e + Cminus 12 Ax

e + g minus C1u( 1113857dt

x2 d

(23)

en the disturbance dynamic system for the spacecraftformation flying is given as

_x1 x2

_x2 _d(24)

For each disturbance di in the spacecraft formationflying system the disturbancersquos model is given as follows

_x1i x2i

_x2i _di i 1 2 6(25)

e supertwisting disturbance observer is given asfollows

_1113957x1i minus λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + 1113957x2i

_1113957x2i minus λ2sign e1i( 1113857(26)

where λ1 and λ2 denote the positive observer gains 1113957x1i and1113957x2i are the estimations of the state vectors x1i and x2i ande1i 1113957x1i minus x1i and e2i 1113957x2i minus x2i are the estimation errorsrespectively

According to the attitude-orbit coupling system (15) and(16) and the supertwisting state observer (26) the estimationerrors can be rewritten as

_e1i minus λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e2i

_e2i minus λ2sign e1i( 1113857 minus d

i(27)

Define the state vectors κ [κ1 κ2]T where

κ1 |e1i|12sign(e1i) κ2 e2i

en the derivatives of the state vectors κ1 and κ2 can bewritten as

_κ1 minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i +12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e2i

_κ2 minus λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i minus d

i

(28)

Similar to [46] the following Lyapunov function for (28)is constructed

Vst κTθ1κ

12

λ21 + 4λ21113872 1113873κ21 minus λ1κ1κ2 + κ22(29)

where θ1 12 (λ21 + 4λ2) minus λ1minus λ1 2

1113890 1113891

en the derivative of Lyapunov function Vst withrespect to the time is given as

_Vst λ21 + 4λ21113872 1113873κ1κ

1 minus λ1κ

1κ2 minus λ1κ1κ

2 + 2κ2κ

2

λ21 + 4λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i +12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e2i1113888 1113889

+λ212

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e22i

minus λ1 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1i minus λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1i minus d

i1113874 1113875

minus 2λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minus 2e2id

i

λ21 + 4λ21113872 1113873 minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

+12

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i1113888 1113889

+λ212

e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minusλ12

e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e22i

+ λ1λ2 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

+ λ1 e1i

11138681113868111386811138681113868111386811138681113868minus (12)

e1id

i

minus 2λ2 e1i

11138681113868111386811138681113868111386811138681113868minus 1

e1ie2i minus 2e2id

i

(30)

Furthermore (30) can be written as follows

Complexity 5

V

st minusλ312

+ λ1λ21113888 1113889 e1i

1113868111386811138681113868111386811138681113868111386812

+ λ21e2isign e1i( 1113857 minusλ12

e22i

e1i

1113868111386811138681113868111386811138681113868111386812

+ λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 _di minus 2e2i

_di

minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868 minus 2λ1e2i e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e

22i1113876 1113877

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2 e1i

11138681113868111386811138681113868111386811138681113868sign e1i( 1113857 minus

4λ1

e1i

1113868111386811138681113868111386811138681113868111386812

e2i1113890 1113891 _di

le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868 minus 2λ1e2i e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e

22i1113876 1113877

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2 e1i

11138681113868111386811138681113868111386811138681113868 +

4λ1

e1i

1113868111386811138681113868111386811138681113868111386812

e2i

111386811138681113868111386811138681113868111386811138681113888 1113889 _di

11138681113868111386811138681113868

11138681113868111386811138681113868

(31)

Since κ21 minus 2|κ1||κ2| + κ22 ge 0 (31) can be rewritten as

V

st le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873κ21 minus 2λ1κ1κ2 + κ221113960 1113961

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2κ21 +

4λ1

κ11113868111386811138681113868

1113868111386811138681113868 κ21113868111386811138681113868

11138681113868111386811138681113888 1113889ςi

le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873κ21 minus 2λ1κ1κ2 + κ221113960 1113961

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2κ21 +

2λ1κ21 +

2λ1κ221113888 1113889ςi

(32)

Define the following matrix as

θ2

λ21 + 2λ2 minus2 λ1 + 1( 1113857ςi

λ1minus λ1

minus λ1 1 minus2ςi

λ1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

en (32) can be concluded as

V

st le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812κ

Tθ2κ (34)

From (33) by selecting appropriate observer gains thematrix θ2 can be a positive definite matrixen (34) becomes

V

st le minusλ1λmin θ2( 1113857

2 e1i

1113868111386811138681113868111386811138681113868111386812 κ

2 (35)

where λmin(θ2) is the minimum eigenvalue of θ2θ1 must to be a positive definite matrix to be Lyapunov

matrix hence the observer gains λ1 and λ2 need to be se-lected appropriately so that θ1 gt 0 Let us define the minimaland maximal eigenvalues of θ1 as λmin(θ1)gt 0 andλmax(θ1)gt 0 respectively en we have

λmin θ1( 1113857κ2 leVst le λmax θ1( 1113857κ

2 (36)

Furthermore by observing from the state vectors κ andκ1 the following can be concluded

e1i

1113868111386811138681113868111386811138681113868111386812

κ211113969

le κ (37)

Finally combining (36) with (37) (35) can be derived as

V

st + λ0V12st le 0

λ0 λ1λmin θ1( 1113857

12λmin θ2( 1113857

2λmax θ1( 1113857gt 0

(38)

Based on (37) and Lemma 1 we prove that limt⟶tde1i 0

and limt⟶tde2i 0 with i 1 2 6 Furthermore the

estimations errors of the state vector x1i and x2i can convergein finite time and the convergence time is td 2V12

st λ0

Remark 2 To guarantee that the trajectory errors convergeto zero within finite time the supertwisting disturbanceobserver gains λ1 and λ2 should be selected appropriately sothat θ1 and θ2 are positive definite matrices

312 Fast Nonsingular Terminal Sliding Mode Control Lawwith Supertwisting Disturbance Observer To design a fast

Fast nonsingular terminalsliding mode control law

based on the supertwistingdisturbance observer

Supertwistingdisturbance observer

Spacecraftformation flying

system model

u

d

Desired statesσd ωd ρcd

xe xde

Figure 2 e block diagram of the control law based on the supertwisting disturbance observer

6 Complexity

nonsingular terminal sliding mode control law the fol-lowing sliding mode surface is chosen

s xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 (39)

where isin1 isin2 gt 0 c1 gt nm and 1lt nmlt 2 are all positiveconstants

If the spacecraft formation flying system state errorsreach the sliding mode s 0 then

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 0 (40)

From (15) substituting the equation x

e xde into (40)we have

x

e minus1isin2

1113888 1113889

mn

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 11138571113872 1113873

mn

minus1isin2

1113888 1113889

mn

xmne 1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d1113872 1113873

mn

(41)

In order to analyze the fast convergence of the non-singular fast terminal sliding mode surface assume that theintegration time of the state variable xe denotes as xe(ti) andxe(ti + ts) respectively en integrating the time alongboth sides of (41)

1113946xe ti+ts( )

xe ti( )

1xe

1113888 1113889

mn

dxe minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1 minus 1

1113872 1113873mn

dτ

le minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

dτ

(42)

Furthermore it can be seen from (42) that the trackingerrors of the spacecraft formation flying system can convergeto zero in a limited time ts le (isinmn

2 (n minus m))xe(ti)1minus (mn)

Remark 3 e proposed nonsingular fast terminal slidingmode surface (39) indicates that when the system statevariable xe is far from the equilibrium point the term xe

plays the major role in making the system trajectoryconverge at a fast rate When xe approaches to theequilibrium point the term isin1|xe|

c1d sign(xe) plays the

dominant role to achieve rapid convergence of the statetrajectory

A design procedure for a fast nonsingular terminalsliding mode control law with a supertwisting disturbanceobserver is given in the following theorem

Theorem 1 Consider the spacecraft formation flying systemdescribed by (15) and (16) with

u Cminus 11 Axde + g minus C21113957x2( 1113857 minus C

minus 11 M u0 + u1 + u2( 1113857 (43)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (44)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (45)

u2 α1sign(s) + α2|s|ηsign(s) (46)

and minus α1sign(s) minus α2|s|ηsign(s) is the reaching law with α1 gt 0

α2 gt 0 and ηgt 1 en the proposed control law along withthe supertwisting disturbance observer (26) can guarantee thetracking errors of the spacecraft formation flying systemconverge to zero in finite time

Proof From (39) the derivative of the sliding surface s isgiven as follows

s

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

xde

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

minus Axde minus g(1113872

+ C1u + C2d11138571113857

(47)

Consider the following Lyapunov function

V1(t) 12sTs + 1113944

6

i1V1i (48)

Taking the time derivative of (48) we have

_V1(t) sTs+ 1113944

6

i1

_V1i

sT

xde + isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C211138721113874

middot x2 minus d( 1113857 minus u0 + u1 + u2( 111385711138571113857

minus α0 1113944

6

i1V

121i

(49)

Substituting the proposed control law (44)ndash(46) into(49)

_V1(t) minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857

minus α0 1113944

6

i1V

121i

(50)

Define xdemin min x(nm)minus 1de1 x

(nm)minus 1de61113966 1113967 α1 xdemin

isin2(nm)α1 and α2 xdeminisin2(nm)α2 then

Complexity 7

_V1(t)le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857 minus α0 1113944

6

i1V

121i

le minus α1s minus α2sη+1

+ isin2n

ms xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

x2

minus d minus α0 1113944

6

i1V

121i

(51)

Define αe isin2(nm)|xde|(nm)minus 1d Mminus 1C2x2 minus d and

select αe α1 minus αe ge 0 we have

_V1(t)le minus αes minus α0 1113944

6

i1V

121i (52)

Finally define αm min2

radicαe α01113864 1113865 then (52) can be

rewritten as

_V1(t)le minus αm

12

radic s + 1113944

6

i1V

121i

⎛⎝ ⎞⎠

le minus αm

12s

21113944

6

i1V1i

⎛⎝ ⎞⎠

12

minus αmV1(t)12

(53)

Remark 4 When the system state variable is far from the fastnonsingular terminal sliding mode surface the termα2|s|

ηsign(s) in the reaching law plays the major role On thecontrary the term α1sign(s) in the reaching law dominatesBy choosing different α1 and α2 and selecting appropriateparameters in the fast terminal sliding mode surface theconvergence speed can be controlled Moreover comparedwith the conventional reaching law in [47] the proposedreaching law has faster convergence and better slidingperformance When s 0 and _s 0 the speed of the statevariable approaching the sliding surface is reduced to zerowhich can alleviate the chattering phenomenon

32 Fast Nonsingular Terminal Sliding Mode Control LawBased on the Adaptive Neural Network In this section anadaptive RBF neural network is employed to approximatethe unknown nonlinear A and g in the spacecraft formationflying system (15) and (16) which is not easy to be preciselyestimated in practical situations e block diagram of a fastnonsingular terminal sliding mode control law based on theadaptive neural network is depicted in Figure 3

e structure of RBF neural network with multiple in-puts and multiple outputs is shown in Figure 4

It can be seen that the RBF neural network has threelayers including input layer output layer and hidden layer

Denote x [x1 x2 xn]T as the input and ϕi(x) is theGaussian function of the ith neuron in the hidden layeren the Gaussian function can be expressed as below

ϕi(x) exp minusx minus ci

2

2g2i

⎛⎝ ⎞⎠ i 1 2 6 (54)

where ci is the ith neural radial basis function and gi is thewidth of the ith neural radial basis function In this sectionwe choosefnn to approximate and compensate the unknownnonlinear parts in the spacecraft formation flying systemwhere fnn minus Axde minus g is the unknown nonlinear term Werewrite fnn as follows

fnn WTi ϕi xi( 1113857 + ξi (55)

where Wi denotes the ideal weighting matrix and ξi denotesthe approximation error i 1 2 6

e estimation of fnn can be expressed as follows1113954fnn 1113954W

Ti ϕi xi( 1113857 (56)

where 1113954Wi denotes the estimation of Wi

Theorem 2 Consider the spacecraft formation flying systemdescribed by (15) and (16) with the following control law andthe adaptive updating law of the RBF neural network

u Cminus 11

1113954fnn minus u31113872 1113873 minus Cminus 11 M u0 + u1 + u2( 1113857 (57)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (58)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (59)

u2 α1sign(s) + α2|s|ηsign(s) (60)

u3 M Mminus 1

C1

1sign(s)dm (61)

1113954W

i Πwϕi xi( 1113857ϱi (62)

where ϱi is ith element of the term sTisin2(nm)|xde|(nm)minus 1d Mminus 1

i 1 2 6 and Πw is a positive symmetrical matrix etracking errors of the spacecraft formation flying system willconverge to zero

Proof Define the error of the weighting vector as1113957Wi Wi minus 1113954Wi Consider the following Lyapunov function

V2(t) 12sTs +

12

1113944

n

i1tr 1113957W

T

i Πminus 1w

1113957Wi1113874 1113875 (63)

Taking the derivative of (63) with respect to time andsubstituting the proposed control law (57)ndash(62) and theadaptive updating law (62) then (63) can be converted as

8 Complexity

_V2(t) sTsminus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

sT

minus isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

1sign(s)dm minus Mminus 1

C2d1113872 11138731113874 1113875

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 11113957fnn

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

(64)

Since

sT isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C2

1sign(s)dm + Mminus 1

C2d1113872 11138731113874 1113875

minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873sign si( 1113857dm

+ 11139446

i1siisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873di

le minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113868111386811138681113868

1113868111386811138681113868dm

+ 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2di

11138681113868111386811138681113868111386811138681113868

le 0

(65)

erefore

_V2(t)le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1 1113957fnn + s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WT

i ϕi xi( 1113857 + ξi1113874 1113875

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944n

i1tr 1113957W

T

i ϕi xi( 11138571113874 1113875sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(66)

Since

sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WTi ϕi xi( 11138571113874 1113875

minus 1113944n

i1tr 1113957W

Ti ϕi xi( 11138571113874 1113875s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(67)

(66) can be rewritten as follows

_V2(t)le minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

sTM

minus 11113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

ms

middot xde

11138681113868111386811138681113868111386811138681113868nm(nm)minus 1minus 1d

Mminus 1

1113944

n

i1ξi

(68)

Define ξh isin2(nm)|xde|(nm)minus 1d Mminus 1 1113936

ni1 ξi then we

have_V2(t)le minus ξs (69)

where ξ α1 minus ξherefore it can be concluded that the tracking errors of

the spacecraft formation flying system converge to zero andthe proof is completed

4 Numerical Simulation

In this section the simulation result that shows the effec-tiveness of the supertwisting disturbance observer thenonsingular fast terminal sliding mode control law and theadaptive neurosliding mode control law are presented re-spectively in each section

e parameters in the formation flying system are givenas follows the masses of the target and the chaser spacecraftare mt mc 110 kg the initial orbit parameters of thetarget spacecraft are set as the semimajor axis a 7000 kmthe eccentricity e 001 the orbit inclination i 30deg andthe ascending node is Ω 45deg and apogee and perigee areboth 0deg

e inertial matrix of the chaser spacecraft isJ [3 1 07 1 35 1 05 1 4] kgmiddotm2 and the disturbance tor-que dτ 01[sin(t) minus cos(03t)sin(05t)]T Nm e desiredattitude and position are given as ρcd [2 minus 2 minus 05]Tm andσd 0001[sin(10t) minus cos(10t)sin(10t)]T

41 Supertwisting Disturbance Observer e gains of thesupertwisting disturbance observer are chosen as λ1 2 and

Complexity 9

Fast nonsingular terminalsliding mode control law basedon the adaptive neural network

Spacecraformation flying

system model

u

Desired statesσd ωd ρcd

xe xde

Adaptive updatinglaw

Figure 3 e block diagram of the control law based on the adaptive neural network

x1

x2

xn

h1

h2

hn

y1

y2

yn

Input layer

Hidden layer

Output layer

Figure 4 e structure of RBF neural network

01

0

ndash01

Estim

atio

ner

ror

0 2 4 6 8 10 12 14 16

16

18

18

20

20

Time (s)

001

0

ndash001

(a)

Estim

atio

ner

ror

005

0

ndash005

ndash010 2 4 6 8 10 12 14 16 18 20

Time (s)

16 18 20

01

0

ndash010 05 1

50

ndash5

times10ndash4

(b)

Figure 5 Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbanceobserver

10 Complexity

0 2 4 6 8 10 12 14 16 18 20Time (s)

ndash10

ndash5

0

5

10

15

20

25

x e re

spon

se

16 17 18 19 20ndash1

0

1times10ndash9

Figure 6 e state responses of xe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash20

ndash10

0

10

20

ndash1

0

1times10ndash6

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de re

spon

se

16 17 18 19 20

Figure 7e state responses of xde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash005

0

005

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e resp

onse

Figure 8 e state responses of xe with the nonlinear PD control law (72)

Complexity 11

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

V

st minusλ312

+ λ1λ21113888 1113889 e1i

1113868111386811138681113868111386811138681113868111386812

+ λ21e2isign e1i( 1113857 minusλ12

e22i

e1i

1113868111386811138681113868111386811138681113868111386812

+ λ1 e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 _di minus 2e2i

_di

minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868 minus 2λ1e2i e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e

22i1113876 1113877

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2 e1i

11138681113868111386811138681113868111386811138681113868sign e1i( 1113857 minus

4λ1

e1i

1113868111386811138681113868111386811138681113868111386812

e2i1113890 1113891 _di

le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873 e1i

11138681113868111386811138681113868111386811138681113868 minus 2λ1e2i e1i

1113868111386811138681113868111386811138681113868111386812sign e1i( 1113857 + e

22i1113876 1113877

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2 e1i

11138681113868111386811138681113868111386811138681113868 +

4λ1

e1i

1113868111386811138681113868111386811138681113868111386812

e2i

111386811138681113868111386811138681113868111386811138681113888 1113889 _di

11138681113868111386811138681113868

11138681113868111386811138681113868

(31)

Since κ21 minus 2|κ1||κ2| + κ22 ge 0 (31) can be rewritten as

V

st le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873κ21 minus 2λ1κ1κ2 + κ221113960 1113961

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2κ21 +

4λ1

κ11113868111386811138681113868

1113868111386811138681113868 κ21113868111386811138681113868

11138681113868111386811138681113888 1113889ςi

le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 λ21 + 2λ21113872 1113873κ21 minus 2λ1κ1κ2 + κ221113960 1113961

+λ1

2 e1i

1113868111386811138681113868111386811138681113868111386812 2κ21 +

2λ1κ21 +

2λ1κ221113888 1113889ςi

(32)

Define the following matrix as

θ2

λ21 + 2λ2 minus2 λ1 + 1( 1113857ςi

λ1minus λ1

minus λ1 1 minus2ςi

λ1

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(33)

en (32) can be concluded as

V

st le minusλ1

2 e1i

1113868111386811138681113868111386811138681113868111386812κ

Tθ2κ (34)

From (33) by selecting appropriate observer gains thematrix θ2 can be a positive definite matrixen (34) becomes

V

st le minusλ1λmin θ2( 1113857

2 e1i

1113868111386811138681113868111386811138681113868111386812 κ

2 (35)

where λmin(θ2) is the minimum eigenvalue of θ2θ1 must to be a positive definite matrix to be Lyapunov

matrix hence the observer gains λ1 and λ2 need to be se-lected appropriately so that θ1 gt 0 Let us define the minimaland maximal eigenvalues of θ1 as λmin(θ1)gt 0 andλmax(θ1)gt 0 respectively en we have

λmin θ1( 1113857κ2 leVst le λmax θ1( 1113857κ

2 (36)

Furthermore by observing from the state vectors κ andκ1 the following can be concluded

e1i

1113868111386811138681113868111386811138681113868111386812

κ211113969

le κ (37)

Finally combining (36) with (37) (35) can be derived as

V

st + λ0V12st le 0

λ0 λ1λmin θ1( 1113857

12λmin θ2( 1113857

2λmax θ1( 1113857gt 0

(38)

Based on (37) and Lemma 1 we prove that limt⟶tde1i 0

and limt⟶tde2i 0 with i 1 2 6 Furthermore the

estimations errors of the state vector x1i and x2i can convergein finite time and the convergence time is td 2V12

st λ0

Remark 2 To guarantee that the trajectory errors convergeto zero within finite time the supertwisting disturbanceobserver gains λ1 and λ2 should be selected appropriately sothat θ1 and θ2 are positive definite matrices

312 Fast Nonsingular Terminal Sliding Mode Control Lawwith Supertwisting Disturbance Observer To design a fast

Fast nonsingular terminalsliding mode control law

based on the supertwistingdisturbance observer

Supertwistingdisturbance observer

Spacecraftformation flying

system model

u

d

Desired statesσd ωd ρcd

xe xde

Figure 2 e block diagram of the control law based on the supertwisting disturbance observer

6 Complexity

nonsingular terminal sliding mode control law the fol-lowing sliding mode surface is chosen

s xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 (39)

where isin1 isin2 gt 0 c1 gt nm and 1lt nmlt 2 are all positiveconstants

If the spacecraft formation flying system state errorsreach the sliding mode s 0 then

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 0 (40)

From (15) substituting the equation x

e xde into (40)we have

x

e minus1isin2

1113888 1113889

mn

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 11138571113872 1113873

mn

minus1isin2

1113888 1113889

mn

xmne 1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d1113872 1113873

mn

(41)

In order to analyze the fast convergence of the non-singular fast terminal sliding mode surface assume that theintegration time of the state variable xe denotes as xe(ti) andxe(ti + ts) respectively en integrating the time alongboth sides of (41)

1113946xe ti+ts( )

xe ti( )

1xe

1113888 1113889

mn

dxe minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1 minus 1

1113872 1113873mn

dτ

le minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

dτ

(42)

Furthermore it can be seen from (42) that the trackingerrors of the spacecraft formation flying system can convergeto zero in a limited time ts le (isinmn

2 (n minus m))xe(ti)1minus (mn)

Remark 3 e proposed nonsingular fast terminal slidingmode surface (39) indicates that when the system statevariable xe is far from the equilibrium point the term xe

plays the major role in making the system trajectoryconverge at a fast rate When xe approaches to theequilibrium point the term isin1|xe|

c1d sign(xe) plays the

dominant role to achieve rapid convergence of the statetrajectory

A design procedure for a fast nonsingular terminalsliding mode control law with a supertwisting disturbanceobserver is given in the following theorem

Theorem 1 Consider the spacecraft formation flying systemdescribed by (15) and (16) with

u Cminus 11 Axde + g minus C21113957x2( 1113857 minus C

minus 11 M u0 + u1 + u2( 1113857 (43)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (44)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (45)

u2 α1sign(s) + α2|s|ηsign(s) (46)

and minus α1sign(s) minus α2|s|ηsign(s) is the reaching law with α1 gt 0

α2 gt 0 and ηgt 1 en the proposed control law along withthe supertwisting disturbance observer (26) can guarantee thetracking errors of the spacecraft formation flying systemconverge to zero in finite time

Proof From (39) the derivative of the sliding surface s isgiven as follows

s

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

xde

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

minus Axde minus g(1113872

+ C1u + C2d11138571113857

(47)

Consider the following Lyapunov function

V1(t) 12sTs + 1113944

6

i1V1i (48)

Taking the time derivative of (48) we have

_V1(t) sTs+ 1113944

6

i1

_V1i

sT

xde + isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C211138721113874

middot x2 minus d( 1113857 minus u0 + u1 + u2( 111385711138571113857

minus α0 1113944

6

i1V

121i

(49)

Substituting the proposed control law (44)ndash(46) into(49)

_V1(t) minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857

minus α0 1113944

6

i1V

121i

(50)

Define xdemin min x(nm)minus 1de1 x

(nm)minus 1de61113966 1113967 α1 xdemin

isin2(nm)α1 and α2 xdeminisin2(nm)α2 then

Complexity 7

_V1(t)le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857 minus α0 1113944

6

i1V

121i

le minus α1s minus α2sη+1

+ isin2n

ms xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

x2

minus d minus α0 1113944

6

i1V

121i

(51)

Define αe isin2(nm)|xde|(nm)minus 1d Mminus 1C2x2 minus d and

select αe α1 minus αe ge 0 we have

_V1(t)le minus αes minus α0 1113944

6

i1V

121i (52)

Finally define αm min2

radicαe α01113864 1113865 then (52) can be

rewritten as

_V1(t)le minus αm

12

radic s + 1113944

6

i1V

121i

⎛⎝ ⎞⎠

le minus αm

12s

21113944

6

i1V1i

⎛⎝ ⎞⎠

12

minus αmV1(t)12

(53)

Remark 4 When the system state variable is far from the fastnonsingular terminal sliding mode surface the termα2|s|

ηsign(s) in the reaching law plays the major role On thecontrary the term α1sign(s) in the reaching law dominatesBy choosing different α1 and α2 and selecting appropriateparameters in the fast terminal sliding mode surface theconvergence speed can be controlled Moreover comparedwith the conventional reaching law in [47] the proposedreaching law has faster convergence and better slidingperformance When s 0 and _s 0 the speed of the statevariable approaching the sliding surface is reduced to zerowhich can alleviate the chattering phenomenon

32 Fast Nonsingular Terminal Sliding Mode Control LawBased on the Adaptive Neural Network In this section anadaptive RBF neural network is employed to approximatethe unknown nonlinear A and g in the spacecraft formationflying system (15) and (16) which is not easy to be preciselyestimated in practical situations e block diagram of a fastnonsingular terminal sliding mode control law based on theadaptive neural network is depicted in Figure 3

e structure of RBF neural network with multiple in-puts and multiple outputs is shown in Figure 4

It can be seen that the RBF neural network has threelayers including input layer output layer and hidden layer

Denote x [x1 x2 xn]T as the input and ϕi(x) is theGaussian function of the ith neuron in the hidden layeren the Gaussian function can be expressed as below

ϕi(x) exp minusx minus ci

2

2g2i

⎛⎝ ⎞⎠ i 1 2 6 (54)

where ci is the ith neural radial basis function and gi is thewidth of the ith neural radial basis function In this sectionwe choosefnn to approximate and compensate the unknownnonlinear parts in the spacecraft formation flying systemwhere fnn minus Axde minus g is the unknown nonlinear term Werewrite fnn as follows

fnn WTi ϕi xi( 1113857 + ξi (55)

where Wi denotes the ideal weighting matrix and ξi denotesthe approximation error i 1 2 6

e estimation of fnn can be expressed as follows1113954fnn 1113954W

Ti ϕi xi( 1113857 (56)

where 1113954Wi denotes the estimation of Wi

Theorem 2 Consider the spacecraft formation flying systemdescribed by (15) and (16) with the following control law andthe adaptive updating law of the RBF neural network

u Cminus 11

1113954fnn minus u31113872 1113873 minus Cminus 11 M u0 + u1 + u2( 1113857 (57)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (58)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (59)

u2 α1sign(s) + α2|s|ηsign(s) (60)

u3 M Mminus 1

C1

1sign(s)dm (61)

1113954W

i Πwϕi xi( 1113857ϱi (62)

where ϱi is ith element of the term sTisin2(nm)|xde|(nm)minus 1d Mminus 1

i 1 2 6 and Πw is a positive symmetrical matrix etracking errors of the spacecraft formation flying system willconverge to zero

Proof Define the error of the weighting vector as1113957Wi Wi minus 1113954Wi Consider the following Lyapunov function

V2(t) 12sTs +

12

1113944

n

i1tr 1113957W

T

i Πminus 1w

1113957Wi1113874 1113875 (63)

Taking the derivative of (63) with respect to time andsubstituting the proposed control law (57)ndash(62) and theadaptive updating law (62) then (63) can be converted as

8 Complexity

_V2(t) sTsminus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

sT

minus isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

1sign(s)dm minus Mminus 1

C2d1113872 11138731113874 1113875

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 11113957fnn

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

(64)

Since

sT isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C2

1sign(s)dm + Mminus 1

C2d1113872 11138731113874 1113875

minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873sign si( 1113857dm

+ 11139446

i1siisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873di

le minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113868111386811138681113868

1113868111386811138681113868dm

+ 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2di

11138681113868111386811138681113868111386811138681113868

le 0

(65)

erefore

_V2(t)le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1 1113957fnn + s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WT

i ϕi xi( 1113857 + ξi1113874 1113875

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944n

i1tr 1113957W

T

i ϕi xi( 11138571113874 1113875sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(66)

Since

sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WTi ϕi xi( 11138571113874 1113875

minus 1113944n

i1tr 1113957W

Ti ϕi xi( 11138571113874 1113875s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(67)

(66) can be rewritten as follows

_V2(t)le minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

sTM

minus 11113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

ms

middot xde

11138681113868111386811138681113868111386811138681113868nm(nm)minus 1minus 1d

Mminus 1

1113944

n

i1ξi

(68)

Define ξh isin2(nm)|xde|(nm)minus 1d Mminus 1 1113936

ni1 ξi then we

have_V2(t)le minus ξs (69)

where ξ α1 minus ξherefore it can be concluded that the tracking errors of

the spacecraft formation flying system converge to zero andthe proof is completed

4 Numerical Simulation

In this section the simulation result that shows the effec-tiveness of the supertwisting disturbance observer thenonsingular fast terminal sliding mode control law and theadaptive neurosliding mode control law are presented re-spectively in each section

e parameters in the formation flying system are givenas follows the masses of the target and the chaser spacecraftare mt mc 110 kg the initial orbit parameters of thetarget spacecraft are set as the semimajor axis a 7000 kmthe eccentricity e 001 the orbit inclination i 30deg andthe ascending node is Ω 45deg and apogee and perigee areboth 0deg

e inertial matrix of the chaser spacecraft isJ [3 1 07 1 35 1 05 1 4] kgmiddotm2 and the disturbance tor-que dτ 01[sin(t) minus cos(03t)sin(05t)]T Nm e desiredattitude and position are given as ρcd [2 minus 2 minus 05]Tm andσd 0001[sin(10t) minus cos(10t)sin(10t)]T

41 Supertwisting Disturbance Observer e gains of thesupertwisting disturbance observer are chosen as λ1 2 and

Complexity 9

Fast nonsingular terminalsliding mode control law basedon the adaptive neural network

Spacecraformation flying

system model

u

Desired statesσd ωd ρcd

xe xde

Adaptive updatinglaw

Figure 3 e block diagram of the control law based on the adaptive neural network

x1

x2

xn

h1

h2

hn

y1

y2

yn

Input layer

Hidden layer

Output layer

Figure 4 e structure of RBF neural network

01

0

ndash01

Estim

atio

ner

ror

0 2 4 6 8 10 12 14 16

16

18

18

20

20

Time (s)

001

0

ndash001

(a)

Estim

atio

ner

ror

005

0

ndash005

ndash010 2 4 6 8 10 12 14 16 18 20

Time (s)

16 18 20

01

0

ndash010 05 1

50

ndash5

times10ndash4

(b)

Figure 5 Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbanceobserver

10 Complexity

0 2 4 6 8 10 12 14 16 18 20Time (s)

ndash10

ndash5

0

5

10

15

20

25

x e re

spon

se

16 17 18 19 20ndash1

0

1times10ndash9

Figure 6 e state responses of xe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash20

ndash10

0

10

20

ndash1

0

1times10ndash6

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de re

spon

se

16 17 18 19 20

Figure 7e state responses of xde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash005

0

005

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e resp

onse

Figure 8 e state responses of xe with the nonlinear PD control law (72)

Complexity 11

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

nonsingular terminal sliding mode control law the fol-lowing sliding mode surface is chosen

s xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 (39)

where isin1 isin2 gt 0 c1 gt nm and 1lt nmlt 2 are all positiveconstants

If the spacecraft formation flying system state errorsreach the sliding mode s 0 then

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 1113857 + isin2 xde

11138681113868111386811138681113868111386811138681113868nmd

sign xde( 1113857 0 (40)

From (15) substituting the equation x

e xde into (40)we have

x

e minus1isin2

1113888 1113889

mn

xe + isin1 xe

11138681113868111386811138681113868111386811138681113868c1dsign xe( 11138571113872 1113873

mn

minus1isin2

1113888 1113889

mn

xmne 1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d1113872 1113873

mn

(41)

In order to analyze the fast convergence of the non-singular fast terminal sliding mode surface assume that theintegration time of the state variable xe denotes as xe(ti) andxe(ti + ts) respectively en integrating the time alongboth sides of (41)

1113946xe ti+ts( )

xe ti( )

1xe

1113888 1113889

mn

dxe minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

1 + isin1 xe

11138681113868111386811138681113868111386811138681113868c1 minus 1

1113872 1113873mn

dτ

le minus 1113946ti+ts

ti

1isin2

1113888 1113889

mn

dτ

(42)

Furthermore it can be seen from (42) that the trackingerrors of the spacecraft formation flying system can convergeto zero in a limited time ts le (isinmn

2 (n minus m))xe(ti)1minus (mn)

Remark 3 e proposed nonsingular fast terminal slidingmode surface (39) indicates that when the system statevariable xe is far from the equilibrium point the term xe

plays the major role in making the system trajectoryconverge at a fast rate When xe approaches to theequilibrium point the term isin1|xe|

c1d sign(xe) plays the

dominant role to achieve rapid convergence of the statetrajectory

A design procedure for a fast nonsingular terminalsliding mode control law with a supertwisting disturbanceobserver is given in the following theorem

Theorem 1 Consider the spacecraft formation flying systemdescribed by (15) and (16) with

u Cminus 11 Axde + g minus C21113957x2( 1113857 minus C

minus 11 M u0 + u1 + u2( 1113857 (43)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (44)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (45)

u2 α1sign(s) + α2|s|ηsign(s) (46)

and minus α1sign(s) minus α2|s|ηsign(s) is the reaching law with α1 gt 0

α2 gt 0 and ηgt 1 en the proposed control law along withthe supertwisting disturbance observer (26) can guarantee thetracking errors of the spacecraft formation flying systemconverge to zero in finite time

Proof From (39) the derivative of the sliding surface s isgiven as follows

s

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

xde

xe

+ isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

minus Axde minus g(1113872

+ C1u + C2d11138571113857

(47)

Consider the following Lyapunov function

V1(t) 12sTs + 1113944

6

i1V1i (48)

Taking the time derivative of (48) we have

_V1(t) sTs+ 1113944

6

i1

_V1i

sT

xde + isin1c1 xe

11138681113868111386811138681113868111386811138681113868c1minus 1d

xde + isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C211138721113874

middot x2 minus d( 1113857 minus u0 + u1 + u2( 111385711138571113857

minus α0 1113944

6

i1V

121i

(49)

Substituting the proposed control law (44)ndash(46) into(49)

_V1(t) minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857

minus α0 1113944

6

i1V

121i

(50)

Define xdemin min x(nm)minus 1de1 x

(nm)minus 1de61113966 1113967 α1 xdemin

isin2(nm)α1 and α2 xdeminisin2(nm)α2 then

Complexity 7

_V1(t)le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857 minus α0 1113944

6

i1V

121i

le minus α1s minus α2sη+1

+ isin2n

ms xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

x2

minus d minus α0 1113944

6

i1V

121i

(51)

Define αe isin2(nm)|xde|(nm)minus 1d Mminus 1C2x2 minus d and

select αe α1 minus αe ge 0 we have

_V1(t)le minus αes minus α0 1113944

6

i1V

121i (52)

Finally define αm min2

radicαe α01113864 1113865 then (52) can be

rewritten as

_V1(t)le minus αm

12

radic s + 1113944

6

i1V

121i

⎛⎝ ⎞⎠

le minus αm

12s

21113944

6

i1V1i

⎛⎝ ⎞⎠

12

minus αmV1(t)12

(53)

Remark 4 When the system state variable is far from the fastnonsingular terminal sliding mode surface the termα2|s|

ηsign(s) in the reaching law plays the major role On thecontrary the term α1sign(s) in the reaching law dominatesBy choosing different α1 and α2 and selecting appropriateparameters in the fast terminal sliding mode surface theconvergence speed can be controlled Moreover comparedwith the conventional reaching law in [47] the proposedreaching law has faster convergence and better slidingperformance When s 0 and _s 0 the speed of the statevariable approaching the sliding surface is reduced to zerowhich can alleviate the chattering phenomenon

32 Fast Nonsingular Terminal Sliding Mode Control LawBased on the Adaptive Neural Network In this section anadaptive RBF neural network is employed to approximatethe unknown nonlinear A and g in the spacecraft formationflying system (15) and (16) which is not easy to be preciselyestimated in practical situations e block diagram of a fastnonsingular terminal sliding mode control law based on theadaptive neural network is depicted in Figure 3

e structure of RBF neural network with multiple in-puts and multiple outputs is shown in Figure 4

It can be seen that the RBF neural network has threelayers including input layer output layer and hidden layer

Denote x [x1 x2 xn]T as the input and ϕi(x) is theGaussian function of the ith neuron in the hidden layeren the Gaussian function can be expressed as below

ϕi(x) exp minusx minus ci

2

2g2i

⎛⎝ ⎞⎠ i 1 2 6 (54)

where ci is the ith neural radial basis function and gi is thewidth of the ith neural radial basis function In this sectionwe choosefnn to approximate and compensate the unknownnonlinear parts in the spacecraft formation flying systemwhere fnn minus Axde minus g is the unknown nonlinear term Werewrite fnn as follows

fnn WTi ϕi xi( 1113857 + ξi (55)

where Wi denotes the ideal weighting matrix and ξi denotesthe approximation error i 1 2 6

e estimation of fnn can be expressed as follows1113954fnn 1113954W

Ti ϕi xi( 1113857 (56)

where 1113954Wi denotes the estimation of Wi

Theorem 2 Consider the spacecraft formation flying systemdescribed by (15) and (16) with the following control law andthe adaptive updating law of the RBF neural network

u Cminus 11

1113954fnn minus u31113872 1113873 minus Cminus 11 M u0 + u1 + u2( 1113857 (57)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (58)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (59)

u2 α1sign(s) + α2|s|ηsign(s) (60)

u3 M Mminus 1

C1

1sign(s)dm (61)

1113954W

i Πwϕi xi( 1113857ϱi (62)

where ϱi is ith element of the term sTisin2(nm)|xde|(nm)minus 1d Mminus 1

i 1 2 6 and Πw is a positive symmetrical matrix etracking errors of the spacecraft formation flying system willconverge to zero

Proof Define the error of the weighting vector as1113957Wi Wi minus 1113954Wi Consider the following Lyapunov function

V2(t) 12sTs +

12

1113944

n

i1tr 1113957W

T

i Πminus 1w

1113957Wi1113874 1113875 (63)

Taking the derivative of (63) with respect to time andsubstituting the proposed control law (57)ndash(62) and theadaptive updating law (62) then (63) can be converted as

8 Complexity

_V2(t) sTsminus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

sT

minus isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

1sign(s)dm minus Mminus 1

C2d1113872 11138731113874 1113875

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 11113957fnn

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

(64)

Since

sT isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C2

1sign(s)dm + Mminus 1

C2d1113872 11138731113874 1113875

minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873sign si( 1113857dm

+ 11139446

i1siisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873di

le minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113868111386811138681113868

1113868111386811138681113868dm

+ 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2di

11138681113868111386811138681113868111386811138681113868

le 0

(65)

erefore

_V2(t)le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1 1113957fnn + s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WT

i ϕi xi( 1113857 + ξi1113874 1113875

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944n

i1tr 1113957W

T

i ϕi xi( 11138571113874 1113875sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(66)

Since

sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WTi ϕi xi( 11138571113874 1113875

minus 1113944n

i1tr 1113957W

Ti ϕi xi( 11138571113874 1113875s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(67)

(66) can be rewritten as follows

_V2(t)le minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

sTM

minus 11113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

ms

middot xde

11138681113868111386811138681113868111386811138681113868nm(nm)minus 1minus 1d

Mminus 1

1113944

n

i1ξi

(68)

Define ξh isin2(nm)|xde|(nm)minus 1d Mminus 1 1113936

ni1 ξi then we

have_V2(t)le minus ξs (69)

where ξ α1 minus ξherefore it can be concluded that the tracking errors of

the spacecraft formation flying system converge to zero andthe proof is completed

4 Numerical Simulation

In this section the simulation result that shows the effec-tiveness of the supertwisting disturbance observer thenonsingular fast terminal sliding mode control law and theadaptive neurosliding mode control law are presented re-spectively in each section

e parameters in the formation flying system are givenas follows the masses of the target and the chaser spacecraftare mt mc 110 kg the initial orbit parameters of thetarget spacecraft are set as the semimajor axis a 7000 kmthe eccentricity e 001 the orbit inclination i 30deg andthe ascending node is Ω 45deg and apogee and perigee areboth 0deg

e inertial matrix of the chaser spacecraft isJ [3 1 07 1 35 1 05 1 4] kgmiddotm2 and the disturbance tor-que dτ 01[sin(t) minus cos(03t)sin(05t)]T Nm e desiredattitude and position are given as ρcd [2 minus 2 minus 05]Tm andσd 0001[sin(10t) minus cos(10t)sin(10t)]T

41 Supertwisting Disturbance Observer e gains of thesupertwisting disturbance observer are chosen as λ1 2 and

Complexity 9

Fast nonsingular terminalsliding mode control law basedon the adaptive neural network

Spacecraformation flying

system model

u

Desired statesσd ωd ρcd

xe xde

Adaptive updatinglaw

Figure 3 e block diagram of the control law based on the adaptive neural network

x1

x2

xn

h1

h2

hn

y1

y2

yn

Input layer

Hidden layer

Output layer

Figure 4 e structure of RBF neural network

01

0

ndash01

Estim

atio

ner

ror

0 2 4 6 8 10 12 14 16

16

18

18

20

20

Time (s)

001

0

ndash001

(a)

Estim

atio

ner

ror

005

0

ndash005

ndash010 2 4 6 8 10 12 14 16 18 20

Time (s)

16 18 20

01

0

ndash010 05 1

50

ndash5

times10ndash4

(b)

Figure 5 Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbanceobserver

10 Complexity

0 2 4 6 8 10 12 14 16 18 20Time (s)

ndash10

ndash5

0

5

10

15

20

25

x e re

spon

se

16 17 18 19 20ndash1

0

1times10ndash9

Figure 6 e state responses of xe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash20

ndash10

0

10

20

ndash1

0

1times10ndash6

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de re

spon

se

16 17 18 19 20

Figure 7e state responses of xde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash005

0

005

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e resp

onse

Figure 8 e state responses of xe with the nonlinear PD control law (72)

Complexity 11

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

_V1(t)le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2 x2 minus d( 1113857 minus α0 1113944

6

i1V

121i

le minus α1s minus α2sη+1

+ isin2n

ms xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

x2

minus d minus α0 1113944

6

i1V

121i

(51)

Define αe isin2(nm)|xde|(nm)minus 1d Mminus 1C2x2 minus d and

select αe α1 minus αe ge 0 we have

_V1(t)le minus αes minus α0 1113944

6

i1V

121i (52)

Finally define αm min2

radicαe α01113864 1113865 then (52) can be

rewritten as

_V1(t)le minus αm

12

radic s + 1113944

6

i1V

121i

⎛⎝ ⎞⎠

le minus αm

12s

21113944

6

i1V1i

⎛⎝ ⎞⎠

12

minus αmV1(t)12

(53)

Remark 4 When the system state variable is far from the fastnonsingular terminal sliding mode surface the termα2|s|

ηsign(s) in the reaching law plays the major role On thecontrary the term α1sign(s) in the reaching law dominatesBy choosing different α1 and α2 and selecting appropriateparameters in the fast terminal sliding mode surface theconvergence speed can be controlled Moreover comparedwith the conventional reaching law in [47] the proposedreaching law has faster convergence and better slidingperformance When s 0 and _s 0 the speed of the statevariable approaching the sliding surface is reduced to zerowhich can alleviate the chattering phenomenon

32 Fast Nonsingular Terminal Sliding Mode Control LawBased on the Adaptive Neural Network In this section anadaptive RBF neural network is employed to approximatethe unknown nonlinear A and g in the spacecraft formationflying system (15) and (16) which is not easy to be preciselyestimated in practical situations e block diagram of a fastnonsingular terminal sliding mode control law based on theadaptive neural network is depicted in Figure 3

e structure of RBF neural network with multiple in-puts and multiple outputs is shown in Figure 4

It can be seen that the RBF neural network has threelayers including input layer output layer and hidden layer

Denote x [x1 x2 xn]T as the input and ϕi(x) is theGaussian function of the ith neuron in the hidden layeren the Gaussian function can be expressed as below

ϕi(x) exp minusx minus ci

2

2g2i

⎛⎝ ⎞⎠ i 1 2 6 (54)

where ci is the ith neural radial basis function and gi is thewidth of the ith neural radial basis function In this sectionwe choosefnn to approximate and compensate the unknownnonlinear parts in the spacecraft formation flying systemwhere fnn minus Axde minus g is the unknown nonlinear term Werewrite fnn as follows

fnn WTi ϕi xi( 1113857 + ξi (55)

where Wi denotes the ideal weighting matrix and ξi denotesthe approximation error i 1 2 6

e estimation of fnn can be expressed as follows1113954fnn 1113954W

Ti ϕi xi( 1113857 (56)

where 1113954Wi denotes the estimation of Wi

Theorem 2 Consider the spacecraft formation flying systemdescribed by (15) and (16) with the following control law andthe adaptive updating law of the RBF neural network

u Cminus 11

1113954fnn minus u31113872 1113873 minus Cminus 11 M u0 + u1 + u2( 1113857 (57)

where

u0 1isin2

m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dsign xde( 1113857 (58)

u1 c1isin1isin2

1113888 1113889m

nxde

111386811138681113868111386811138681113868111386811138682minus (nm)

dxe

11138681113868111386811138681113868111386811138681113868c1minus 1d

sign xde( 1113857 (59)

u2 α1sign(s) + α2|s|ηsign(s) (60)

u3 M Mminus 1

C1

1sign(s)dm (61)

1113954W

i Πwϕi xi( 1113857ϱi (62)

where ϱi is ith element of the term sTisin2(nm)|xde|(nm)minus 1d Mminus 1

i 1 2 6 and Πw is a positive symmetrical matrix etracking errors of the spacecraft formation flying system willconverge to zero

Proof Define the error of the weighting vector as1113957Wi Wi minus 1113954Wi Consider the following Lyapunov function

V2(t) 12sTs +

12

1113944

n

i1tr 1113957W

T

i Πminus 1w

1113957Wi1113874 1113875 (63)

Taking the derivative of (63) with respect to time andsubstituting the proposed control law (57)ndash(62) and theadaptive updating law (62) then (63) can be converted as

8 Complexity

_V2(t) sTsminus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

sT

minus isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

1sign(s)dm minus Mminus 1

C2d1113872 11138731113874 1113875

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 11113957fnn

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

(64)

Since

sT isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C2

1sign(s)dm + Mminus 1

C2d1113872 11138731113874 1113875

minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873sign si( 1113857dm

+ 11139446

i1siisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873di

le minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113868111386811138681113868

1113868111386811138681113868dm

+ 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2di

11138681113868111386811138681113868111386811138681113868

le 0

(65)

erefore

_V2(t)le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1 1113957fnn + s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WT

i ϕi xi( 1113857 + ξi1113874 1113875

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944n

i1tr 1113957W

T

i ϕi xi( 11138571113874 1113875sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(66)

Since

sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WTi ϕi xi( 11138571113874 1113875

minus 1113944n

i1tr 1113957W

Ti ϕi xi( 11138571113874 1113875s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(67)

(66) can be rewritten as follows

_V2(t)le minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

sTM

minus 11113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

ms

middot xde

11138681113868111386811138681113868111386811138681113868nm(nm)minus 1minus 1d

Mminus 1

1113944

n

i1ξi

(68)

Define ξh isin2(nm)|xde|(nm)minus 1d Mminus 1 1113936

ni1 ξi then we

have_V2(t)le minus ξs (69)

where ξ α1 minus ξherefore it can be concluded that the tracking errors of

the spacecraft formation flying system converge to zero andthe proof is completed

4 Numerical Simulation

In this section the simulation result that shows the effec-tiveness of the supertwisting disturbance observer thenonsingular fast terminal sliding mode control law and theadaptive neurosliding mode control law are presented re-spectively in each section

e parameters in the formation flying system are givenas follows the masses of the target and the chaser spacecraftare mt mc 110 kg the initial orbit parameters of thetarget spacecraft are set as the semimajor axis a 7000 kmthe eccentricity e 001 the orbit inclination i 30deg andthe ascending node is Ω 45deg and apogee and perigee areboth 0deg

e inertial matrix of the chaser spacecraft isJ [3 1 07 1 35 1 05 1 4] kgmiddotm2 and the disturbance tor-que dτ 01[sin(t) minus cos(03t)sin(05t)]T Nm e desiredattitude and position are given as ρcd [2 minus 2 minus 05]Tm andσd 0001[sin(10t) minus cos(10t)sin(10t)]T

41 Supertwisting Disturbance Observer e gains of thesupertwisting disturbance observer are chosen as λ1 2 and

Complexity 9

Fast nonsingular terminalsliding mode control law basedon the adaptive neural network

Spacecraformation flying

system model

u

Desired statesσd ωd ρcd

xe xde

Adaptive updatinglaw

Figure 3 e block diagram of the control law based on the adaptive neural network

x1

x2

xn

h1

h2

hn

y1

y2

yn

Input layer

Hidden layer

Output layer

Figure 4 e structure of RBF neural network

01

0

ndash01

Estim

atio

ner

ror

0 2 4 6 8 10 12 14 16

16

18

18

20

20

Time (s)

001

0

ndash001

(a)

Estim

atio

ner

ror

005

0

ndash005

ndash010 2 4 6 8 10 12 14 16 18 20

Time (s)

16 18 20

01

0

ndash010 05 1

50

ndash5

times10ndash4

(b)

Figure 5 Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbanceobserver

10 Complexity

0 2 4 6 8 10 12 14 16 18 20Time (s)

ndash10

ndash5

0

5

10

15

20

25

x e re

spon

se

16 17 18 19 20ndash1

0

1times10ndash9

Figure 6 e state responses of xe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash20

ndash10

0

10

20

ndash1

0

1times10ndash6

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de re

spon

se

16 17 18 19 20

Figure 7e state responses of xde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash005

0

005

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e resp

onse

Figure 8 e state responses of xe with the nonlinear PD control law (72)

Complexity 11

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

_V2(t) sTsminus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

sT

minus isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2

1sign(s)dm minus Mminus 1

C2d1113872 11138731113874 1113875

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 11113957fnn

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

(64)

Since

sT isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus Mminus 1

C2

1sign(s)dm + Mminus 1

C2d1113872 11138731113874 1113875

minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873sign si( 1113857dm

+ 11139446

i1siisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113872 1113873di

le minus 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C21113868111386811138681113868

1113868111386811138681113868dm

+ 11139446

i1si

11138681113868111386811138681113868111386811138681113868isin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

C2di

11138681113868111386811138681113868111386811138681113868

le 0

(65)

erefore

_V2(t)le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1 1113957fnn + s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944

n

i1tr 1113957W

T

i Πminus 1w

_1113957Wi1113874 1113875

le sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WT

i ϕi xi( 1113857 + ξi1113874 1113875

+ sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

minus α1sign(s) minus α2|s|ηsign(s)( 1113857

minus 1113944n

i1tr 1113957W

T

i ϕi xi( 11138571113874 1113875sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(66)

Since

sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1

1113957WTi ϕi xi( 11138571113874 1113875

minus 1113944n

i1tr 1113957W

Ti ϕi xi( 11138571113874 1113875s

Tisin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

(67)

(66) can be rewritten as follows

_V2(t)le minus sTisin2

n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

α1sign(s) + α2|s|ηsign(s)( 1113857

+ isin2n

mxde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

sTM

minus 11113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

msT

xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

middot xde

11138681113868111386811138681113868111386811138681113868(nm)minus 1d

Mminus 1

1113944

n

i1ξi

le minus α1s minus α2sη+1

+ isin2n

ms

middot xde

11138681113868111386811138681113868111386811138681113868nm(nm)minus 1minus 1d

Mminus 1

1113944

n

i1ξi

(68)

Define ξh isin2(nm)|xde|(nm)minus 1d Mminus 1 1113936

ni1 ξi then we

have_V2(t)le minus ξs (69)

where ξ α1 minus ξherefore it can be concluded that the tracking errors of

the spacecraft formation flying system converge to zero andthe proof is completed

4 Numerical Simulation

In this section the simulation result that shows the effec-tiveness of the supertwisting disturbance observer thenonsingular fast terminal sliding mode control law and theadaptive neurosliding mode control law are presented re-spectively in each section

e parameters in the formation flying system are givenas follows the masses of the target and the chaser spacecraftare mt mc 110 kg the initial orbit parameters of thetarget spacecraft are set as the semimajor axis a 7000 kmthe eccentricity e 001 the orbit inclination i 30deg andthe ascending node is Ω 45deg and apogee and perigee areboth 0deg

e inertial matrix of the chaser spacecraft isJ [3 1 07 1 35 1 05 1 4] kgmiddotm2 and the disturbance tor-que dτ 01[sin(t) minus cos(03t)sin(05t)]T Nm e desiredattitude and position are given as ρcd [2 minus 2 minus 05]Tm andσd 0001[sin(10t) minus cos(10t)sin(10t)]T

41 Supertwisting Disturbance Observer e gains of thesupertwisting disturbance observer are chosen as λ1 2 and

Complexity 9

Fast nonsingular terminalsliding mode control law basedon the adaptive neural network

Spacecraformation flying

system model

u

Desired statesσd ωd ρcd

xe xde

Adaptive updatinglaw

Figure 3 e block diagram of the control law based on the adaptive neural network

x1

x2

xn

h1

h2

hn

y1

y2

yn

Input layer

Hidden layer

Output layer

Figure 4 e structure of RBF neural network

01

0

ndash01

Estim

atio

ner

ror

0 2 4 6 8 10 12 14 16

16

18

18

20

20

Time (s)

001

0

ndash001

(a)

Estim

atio

ner

ror

005

0

ndash005

ndash010 2 4 6 8 10 12 14 16 18 20

Time (s)

16 18 20

01

0

ndash010 05 1

50

ndash5

times10ndash4

(b)

Figure 5 Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbanceobserver

10 Complexity

0 2 4 6 8 10 12 14 16 18 20Time (s)

ndash10

ndash5

0

5

10

15

20

25

x e re

spon

se

16 17 18 19 20ndash1

0

1times10ndash9

Figure 6 e state responses of xe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash20

ndash10

0

10

20

ndash1

0

1times10ndash6

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de re

spon

se

16 17 18 19 20

Figure 7e state responses of xde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash005

0

005

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e resp

onse

Figure 8 e state responses of xe with the nonlinear PD control law (72)

Complexity 11

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

Fast nonsingular terminalsliding mode control law basedon the adaptive neural network

Spacecraformation flying

system model

u

Desired statesσd ωd ρcd

xe xde

Adaptive updatinglaw

Figure 3 e block diagram of the control law based on the adaptive neural network

x1

x2

xn

h1

h2

hn

y1

y2

yn

Input layer

Hidden layer

Output layer

Figure 4 e structure of RBF neural network

01

0

ndash01

Estim

atio

ner

ror

0 2 4 6 8 10 12 14 16

16

18

18

20

20

Time (s)

001

0

ndash001

(a)

Estim

atio

ner

ror

005

0

ndash005

ndash010 2 4 6 8 10 12 14 16 18 20

Time (s)

16 18 20

01

0

ndash010 05 1

50

ndash5

times10ndash4

(b)

Figure 5 Comparisons of disturbance estimation error between (a) the linear disturbance observer and (b) the supertwisting disturbanceobserver

10 Complexity

0 2 4 6 8 10 12 14 16 18 20Time (s)

ndash10

ndash5

0

5

10

15

20

25

x e re

spon

se

16 17 18 19 20ndash1

0

1times10ndash9

Figure 6 e state responses of xe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash20

ndash10

0

10

20

ndash1

0

1times10ndash6

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de re

spon

se

16 17 18 19 20

Figure 7e state responses of xde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash005

0

005

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e resp

onse

Figure 8 e state responses of xe with the nonlinear PD control law (72)

Complexity 11

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

0 2 4 6 8 10 12 14 16 18 20Time (s)

ndash10

ndash5

0

5

10

15

20

25

x e re

spon

se

16 17 18 19 20ndash1

0

1times10ndash9

Figure 6 e state responses of xe with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash20

ndash10

0

10

20

ndash1

0

1times10ndash6

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de re

spon

se

16 17 18 19 20

Figure 7e state responses of xde with the fast nonsingular terminal sliding mode control law (43) based on the supertwisting disturbanceobserver

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash005

0

005

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e resp

onse

Figure 8 e state responses of xe with the nonlinear PD control law (72)

Complexity 11

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

ndash80

ndash60

ndash40

ndash20

0

20

40

ndash05

0

05

0 2 4 6 8 10 12 14 16 18 20Time (s)

x de r

espo

nse

16 17 18 19 20

Figure 9 e state responses of xde with the nonlinear PD control law (72)

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 10 e state responses of xe with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

ndash10

ndash5

0

5

10

15

20

25

16 18 20ndash01

0

01

0 2 4 6 8 10 12 14 16 18 20Time (s)

x e re

spon

se

Figure 11 e state responses of xde with the fast nonsingular terminal sliding mode control law (57) based on the adaptive RBF neuralnetwork

12 Complexity

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

λ2 4 In order to demonstrate the advantage of the pro-posed supertwisting disturbance observer we compare itsperformance with the traditional linear observer as below

_1113954x1 minus L1 x1 minus 1113954x1( 1113857 + x2

_1113954x2 minus L2 x2 minus 1113954x2( 1113857

⎧⎨

⎩ (70)

e L1 and L2 in (70) are selected as

L1 diag 0023 0011 0014 0012 0010 0008 times 104

L2 diag 1320 0300 0400 0270 0160 0070 times 104(71)

e disturbance estimation error given by the lineardisturbance observer (70) and the proposed supertwistingdisturbance observer (26) are depicted in Figures 5(a) and5(b) respectively

Remark 5 It can be seen from Figure 5 that the super-twisting disturbance observer estimation error converges tozero in finite time whereas the estimation error from thelinear disturbance observer takes longer time to converge tozero erefore the proposed supertwisting observer canestimate the disturbance in finite time which demonstratesits advantage over the linear one in the spacecraft formationflying system

42eNonsingular Fast Terminal SlidingMode Control LawwithSupertwistingDisturbanceObserver eparameters forthe nonsingular fast terminal sliding mode surface are set asisin1 01 isin2 002 c1 2719 and nm 2119 e pa-rameters in the control law are set as α1 100 α2 01isin 05 and η 15

e state responses of xe and xde under the proposedcontrol sliding model control law (43) are given in Figures 6and 7 respectively

Furthermore to illustrate the effectiveness of the pro-posed control law in this part we compare it with a widelyused nonlinear PD control law given as follows

u Cminus 11 M minus Kpxe minus Kdxde1113872 1113873 (72)

where the coefficient Kp and Kd are calculated as follows

Kp diag 165 375 50 135 20 875

Kd diag 575 275 35 30 25 20 (73)

e state responses of vectors xe and xde under thenonlinear PD control law are given in Figures 8 and 9

Remark 6 From Figures 6ndash9 it can be seen that the pro-posed control law (43) yields better transient responses ascompared to the nonlinear PD control law Also it is obviousthat the proposed control law (43) guarantees the trackingerrors converge to zero in a finite time

43 Adaptive Neurosliding Mode Control Law e param-eters of the RBF neural network ci is evenly selected over the

space and gi 10 200 neurons are used to approximate eachfnni

Πw diag 100 100 100 100 100 100 100 e state responses of the system under the RBF neural

network control law (57) are given in Figures 10 and 11

Remark 7 Figures 10 and 11 indicate that the trackingerrors converge to a small region of zero with a goodadaptive performance is concludes that the RBF neuralnetwork control law (57) is able to compensate for theunknown dynamics in the spacecraft formation flyingsystem

5 Conclusion

is paper has proposed the nonsingular fast terminalsliding mode control laws to solve the tracking controlproblem for spacecraft formation flying systems First asupertwisting disturbance observer has been introduced toestimate the disturbances in finite time en based on anonsingular fast terminal sliding mode surface a novelcontrol law has been presented to guarantee the trackingerrors of the spacecraft formation converge to zero in finitetime Furthermore for the unknown nonlinear parts in thespacecraft formation flying system a RBF neural networkhas been proposed to approximate them and then com-pensate them Finally via simulations the effectiveness of theproposed control laws has been demonstrated In our futurework we will investigate (1) the spacecraft formation systemsubject to the time-delay on terminal adaptive neuroslidingmode control [48 49] and (2) nonlinear observer design forthe spacecraft formation system

Data Availability

e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

All authors declare that they have no conflicts of interest

Acknowledgments

is work was supported by the Scholarship from ChinaScholarship Council

References

[1] K Lu Y Xia and M Fu ldquoController design for rigidspacecraft attitude tracking with actuator saturationrdquo Infor-mation Sciences vol 220 pp 343ndash366 2013

[2] L Sun W Huo and Z Jiao ldquoAdaptive backstepping controlof spacecraft rendezvous and proximity operations with inputsaturation and full-state constraintrdquo IEEE Transactions onIndustrial Electronics vol 64 no 1 pp 480ndash492 2017

[3] H-T Chen S-M Song and Z-B Zhu ldquoRobust finite-timeattitude tracking control of rigid spacecraft under actuatorsaturationrdquo International Journal of Control Automation andSystems vol 16 no 1 pp 1ndash15 2018

[4] Q Li B Zhang J Yuan and H Wang ldquoPotential functionbased robust safety control for spacecraft rendezvous and

Complexity 13

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

proximity operations under path constraintrdquo Advances inSpace Research vol 62 no 9 pp 2586ndash2598 2018

[5] Z Zhu Y Guo and Z Gao ldquoAdaptive finite-time actuatorfault-tolerant coordinated attitude control of multispacecraftwith input saturationrdquo International Journal of AdaptiveControl and Signal Processing vol 33 no 4 pp 644ndash6632019

[6] M-D Tran and H-J Kang ldquoNonsingular terminal slidingmode control of uncertain second-order nonlinear systemsrdquoMathematical Problems in Engineering vol 2015 Article ID181737 8 pages 2015

[7] J Wan T Hayat and F E Alsaadi ldquoAdaptive neural globallyasymptotic tracking control for a class of uncertain nonlinearsystemsrdquo IEEE Access vol 7 pp 19054ndash19062 2019

[8] E Capello E Punta F Dabbene G Guglieri and R TempoldquoSliding-mode control strategies for rendezvous and dockingmaneuversrdquo Journal of Guidance Control and Dynamicsvol 40 no 6 pp 1481ndash1487 2017

[9] C Pukdeboon ldquoAdaptive backstepping finite-time slidingmode control of spacecraft attitude trackingrdquo Journal ofSystems Engineering and Electronics vol 26 pp 826ndash8392015

[10] B Cai L Zhang and Y Shi ldquoControl synthesis of hiddensemi-markov uncertain fuzzy systems via observations ofhidden modesrdquo IEEE Transactions on Cybernetics pp 1ndash102019

[11] X Lin X Shi and S Li ldquoOptimal cooperative control forformation flying spacecraft with collision avoidancerdquo ScienceProgress vol 103 no 1 pp 1ndash9 2020

[12] R Liu M Liu and Y Liu ldquoNonlinear optimal trackingcontrol of spacecraft formation flying with collision avoid-ancerdquo Transactions of the Institute of Measurement andControl vol 41 no 4 pp 889ndash899 2019

[13] X Cao P Shi Z Li and M Liu ldquoNeural-network-basedadaptive backstepping control with application to spacecraftattitude regulationrdquo IEEE Transactions on Neural Networksand Learning Systems vol 29 no 9 pp 4303ndash4313 2018

[14] Y Wang B Jiang Z-G Wu S Xie and Y Peng ldquoAdaptivesliding mode fault-tolerant fuzzy tracking control with ap-plication to unmanned marine vehiclesrdquo IEEE Transactionson Systems Man and Cybernetics Systems pp 1ndash10 2020

[15] J Fei and Z Feng ldquoAdaptive fuzzy super-twisting slidingmode control for microgyroscoperdquo Complexity vol 2019Article ID 6942642 13 pages 2019

[16] J Fei and H Wang ldquoExperimental investigation of recurrentneural network fractional-order sliding mode control foractive power filterrdquo IEEE Transactions on Circuits and SystemsII-Express Briefs 2019

[17] Y Wang W Zhou J Luo H Yan H Pu and Y PengldquoReliable intelligent path following control for a roboticairship against sensor faultsrdquo IEEEASME Transactions onMechatronics vol 24 no 6 pp 2572ndash2582 2019

[18] G-Q Wu and S-M Song ldquoAntisaturation attitude and orbit-coupled control for spacecraft final safe approach based onfast nonsingular terminal sliding moderdquo Journal of AerospaceEngineering vol 32 no 2 2019

[19] D Ran B Xiao T Sheng and X Chen ldquoAdaptive finite timecontrol for spacecraft attitude maneuver based on second-order terminal sliding moderdquo Proceedings of the Institution ofMechanical Engineers Part G Journal of Aerospace Engi-neering vol 231 no 8 pp 1415ndash1427 2017

[20] Z Wang Y Su and L Zhang ldquoA new nonsingular terminalsliding mode control for rigid spacecraft attitude trackingrdquo

Journal Dynamic System Measurement Control vol 140no 5 2018

[21] C Jing H Xu X Niu and X Song ldquoAdaptive nonsingularterminal sliding mode control for attitude tracking ofspacecraft with actuator faultsrdquo IEEE Access vol 7pp 31485ndash31493 2019

[22] L Yang and Y Jia ldquoFinite-time attitude tracking control for arigid spacecraft using time-varying terminal sliding modetechniquesrdquo International Journal of Control vol 88 no 6pp 1150ndash1162 2015

[23] J Fei and Z Feng ldquoFractional-order finite-time super-twisting sliding mode control of micro gyroscope based ondouble-loop fuzzy neural networkrdquo IEEE Transactions onSystems Man and Cybernetics Systems pp 1ndash15 2020

[24] L Tao Q Chen and Y Nan ldquoDisturbance-observer basedadaptive control for second-order nonlinear systems usingchattering-free reaching lawrdquo International Journal of Con-trol Automation and Systems vol 17 no 2 pp 356ndash3692019

[25] L Zhou J She S Zhou and C Li ldquoCompensation for state-dependent nonlinearity in a modified repetitive control sys-temrdquo International Journal of Robust and Nonlinear Controlvol 28 no 1 pp 213ndash226 2018

[26] Q Li J Yuan B Zhang and H Wang ldquoDisturbance observerbased control for spacecraft proximity operations with pathconstraintrdquo Aerospace Science and Technology vol 79pp 154ndash163 2018

[27] W Deng and J Yao ldquoExtended-state-observer-based adaptivecontrol of electro-hydraulic servomechanisms without ve-locity measurementrdquo IEEEASME Transactions on Mecha-tronics p 1 2019

[28] W Deng J Yao and D Ma ldquoTime-varying input delaycompensation for nonlinear systems with additive distur-bance an output feedback approachrdquo International Journal ofRobust and Nonlinear Control vol 28 no 1 pp 31ndash52 2018

[29] A A Godbole J P Kolhe and S E Talole ldquoPerformanceanalysis of generalized extended state observer in tacklingsinusoidal disturbancesrdquo IEEE Transactions on Control Sys-tems Technology vol 21 no 6 pp 2212ndash2223 2013

[30] Q Zong Q Dong F Wang and B Tian ldquoSuper twistingsliding mode control for a flexible air-breathing hypersonicvehicle based on disturbance observerrdquo Science China In-formation Sciences vol 58 no 7 pp 1ndash15 2015

[31] H Yoon Y Eun and C Park ldquoAdaptive tracking control ofspacecraft relative motion with mass and thruster uncer-taintiesrdquoAerospace Science and Technology vol 34 pp 75ndash832014

[32] A-M Zou K D Kumar and Z-G Hou ldquoDistributedconsensus control for multi-agent systems using terminalsliding mode and Chebyshev neural networksrdquo InternationalJournal of Robust and Nonlinear Control vol 23 no 3pp 334ndash357 2013

[33] M Van and H-J Kang ldquoRobust fault-tolerant control foruncertain robot manipulators based on adaptive quasi-con-tinuous high-order sliding mode and neural networkrdquo Pro-ceedings of the Institution of Mechanical Engineers Part CJournal of Mechanical Engineering Science vol 229 no 8pp 1425ndash1446 2015

[34] Y Sun G Ma L Chen and P Wang ldquoNeural network-baseddistributed adaptive configuration containment control forsatellite formationsrdquo Proceedings of the Institution of Me-chanical Engineers Part G Journal of Aerospace Engineeringvol 232 no 12 pp 2349ndash2363 2017

14 Complexity

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15

[35] Y Sun L Chen G Ma and C Li ldquoAdaptive neural networktracking control for multiple uncertain Euler-Lagrange sys-tems with communication delaysrdquo Journal of the FranklinInstitute vol 354 no 7 pp 2677ndash2698 2017

[36] Z Jin J Chen Y Sheng and X Liu ldquoNeural network basedadaptive fuzzy PID-type sliding mode attitude control for areentry vehiclerdquo International Journal of Control Automationand Systems vol 15 no 1 pp 404ndash415 2017

[37] C Yang T Teng B Xu Z Li J Na and C-Y Su ldquoGlobaladaptive tracking control of robot manipulators using neuralnetworks with finite-time learning convergencerdquo Interna-tional Journal of Control Automation and Systems vol 15no 4 pp 1916ndash1924 2017

[38] L Zhao and Y Jia ldquoNeural network-based distributedadaptive attitude synchronization control of spacecraft for-mation under modified fast terminal sliding moderdquo Neuro-computing vol 171 pp 230ndash241 2016

[39] D Sun F Naghdy and H Du ldquoNeural network-basedpassivity control of teleoperation system under time-varyingdelaysrdquo IEEE Transactions on Cybernetics vol 47 no 7pp 1666ndash1680 2017

[40] N Zhou R Chen Y Xia and J Huang ldquoEstimator-basedadaptive neural network control of leader-follower high-ordernonlinear multiagent systems with actuator faultsrdquo Concur-rency and Computation Practice and Experience vol 29no 24 pp 1ndash13 2017

[41] X Gao X Zhong D You and S Katayama ldquoKalman filteringcompensated by radial basis function neural network for seamtracking of laser weldingrdquo IEEE Transactions on ControlSystems Technology vol 21 no 5 pp 1916ndash1923 2013

[42] J Fei and Y Chen ldquoDynamic terminal sliding mode controlfor single-phase active power filter using double hidden layerrecurrent neural networkrdquo IEEE Transactions on PowerElectronics vol 35 no 9 pp 9906ndash9924 2020

[43] Y Hong J Wang and D Cheng ldquoAdaptive finite-timecontrol of nonlinear systems with parametric uncertaintyrdquoIEEE Transactions on Automatic Control vol 51 no 5pp 858ndash862 2006

[44] C Pukdeboon ldquoOutput feedback second order sliding modecontrol for spacecraft attitude and translation motionrdquo In-ternational Journal of Control Automation and Systemsvol 14 no 2 pp 411ndash424 2016

[45] T Chen and H Chen ldquoApproximation capability to functionsof several variables nonlinear functionals and operators byradial basis function neural networksrdquo IEEE TransactionsNeural Network vol 6 no 4 pp 904ndash910 1995

[46] E Duraffourg L Burlion and T Ahmed-Ali ldquoFinite-timeobserver-based backstepping control of a flexible launchvehiclerdquo Journal of Vibration and Control vol 24 no 8pp 1535ndash1550 2018

[47] N Tino P Siricharuanun and C Pukdeboon ldquoChaos syn-chronization of two different chaotic systems via nonsingularterminal sliding mode techniquesrdquo ai Journal of Mathe-matics vol 14 pp 22ndash36 2016

[48] X Ge Q-L Han and X-M Zhang ldquoAchieving clusterformation of multi-agent systems under aperiodic samplingand communication delaysrdquo IEEE Transactions on IndustrialElectronics vol 65 no 4 pp 3417ndash3426 2018

[49] B-L Zhang Q-L Han X-M Zhang and X Yu ldquoSlidingmode control with mixed current and delayed states foroffshore steel jacket platformsrdquo IEEE Transactions on ControlSystems Technology vol 22 pp 1769ndash1783 2014

Complexity 15