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Acta Astronautica

Acta Astronautica 104 (2014) 304–312

http://d0094-57

n CorrE-m

journal homepage: www.elsevier.com/locate/actaastro

Fractional order tension control for stable and fast tetheredsatellite retrieval

Guanghui Sun a,b, Z.H. Zhu b,n

a School of Astronautics, York university, 92 Xidazhi Street, Harbin 150001, Chinab Department of Earth and Space Science and Engineering, York University, 4700 Keele Street, Toronto, Ontario, Canada, M3J 1P3

a r t i c l e i n f o

Article history:Received 7 March 2014Received in revised form11 August 2014Accepted 15 August 2014Available online 28 August 2014

Keywords:Tethered satelliteRetrievalFractional orderTension controlDynamics

x.doi.org/10.1016/j.actaastro.2014.08.01265/& 2014 IAA. Published by Elsevier Ltd. A

esponding author. Tel.: þ1 416 736 2100x77ail address: gzhu@yorku.ca (Z.H. Zhu).

a b s t r a c t

The retrieval of a tethered satellite system is intrinsically unstable. This paper develops anew control strategy to retrieve the tethered satellite system stably and quickly using thefractional order control theory. The governing equation of the tethered satellite systemand classic linear feedback tension control law were first reviewed and examined as abenchmark. Then, a new fractional order tension control law has been to avoid thetethered satellite winds around the main satellite near the end of retrieval by existinginteger order tension control laws. The newly proposed control law has been discretizedand implemented by the Laplace transform and Tustin operator. Unlike the existinginteger order control laws, which are based on the feedback of current state andmemoryless, the fractional order control law has the memory of previous states and thuscontrols the tether retrieval more smoothly while maintaining the retrieving speed. Theeffectiveness and advantage of the new fractional order tension control law is demon-strated numerically by comparing with its integer order counterpart. The results showthat the new control law not only retrieves the subsatellite without winding around themain satellite, but also provides a better control performance with smaller in-planelibration angles.

& 2014 IAA. Published by Elsevier Ltd. All rights reserved.

1. Introduction

The past decades have witnessed 26 tethered satellitemissions [1] due to its great potential in space explora-tion, microgravity and upper atmospheric studies, electro-dynamic propulsion and deep space observation [2–4].A typical tethered satellite mission generally involves tetherdeployment, state keeping, and tether retrieval, where thetether deployment and retrieval are the most critical stages[2,5,6]. The deployment process is generally stable due tothe stabilizing effect of gravity gradient field and can becontrolled by an open-loop and linear-exponential control

ll rights reserved.

729.

law [7]. In contrast, the retrieval process is inherentlyunstable [8], especially as the subsatellite approaches tothe main satellite. Many control methods have been pro-posed to retrieve the tethered satellite in a stable and fastmanner in the past decades, such as tension feedbackcontrol strategy to retrieve the tether with a windlassmechanism [9,10], open-loop control strategy to retrievethe tether uniformly or exponentially [7], optimal controls[10], delayed feedback controls [11], and others [12,13].Among them, the tension control is most appealing due toits simplicity, effectiveness, and implementability [14].

The tension control law was first introduced by Rupp[9], where the tether tension was regulated by a prede-termined control law to retrieve the subsatellite to therequired length while keeping the tether libration as smallas possible. Baker et al. [15] applied the tension control law

x

y

z

Fig. 1. The local orbital reference frame of TSS.

G. Sun, Z.H. Zhu / Acta Astronautica 104 (2014) 304–312 305

to control the out-of-plane libration through the higher-order modal coupling between the in-plane and out-of-plane libration modes. Pradeep [16] developed asimple tension control law based on the linearization ofthe equation of in-plane motion of tether in the neighbor-hood of a system stationary point using the KTC (Kelvin–Talt–Chetayev) theorem for in-plane libration control.Kumar and Pradeep [17] generalized the Pradeep's simpletension control law to three-dimensional tethered satellitesystems. Modi et al. [18] derived a nonlinear feedbackcontrol law that included the out-of-plane tether angularvelocity to eliminate the librations. Liaw and Abed [19]proposed a controller based on the Hopf bifurcationtheorem. Licata [20] studied a fuzzy logic feedback tensioncontrol scheme for the in-plane tether deployment.A detailed review can be found in [21]. Although effective,the existing control schemes cannot achieve the stable andfast retrieval of tethered subsatellite simultaneously.To achieve the balance between the stability and efficiencyof retrieval, a memory feedback control is highly desirableto automatically adjust the control performance based onits time history.

Fractional order calculus possesses the historical mem-ory capacity by generalizing the classical integer orderdifferential and integral calculus to a non-integer ordercombined differentiation/integration operator, called dif-ferintegral. It dates back to the birth of classical integerorder calculus, but received little attention due to theinherent mathematic complexity and lack of clear physicalmeaning. Over the past decades, it has been observed thatmany physical systems can be described more accuratelyby the fractional order calculus than the classical calculusdue to its unique historical memory capacity, for instance,the viscoelastic systems [22], polymeric chemistry systems[23], biomedical systems [24], and signal processing [25],just to name a few. In the field of dynamics and control,Fractional Order Control (FOC) theory attracts great inter-est since it can provide a better and robust performancefor a closed-loop control system. Over the years, manyeffective FOC methods have been proposed, such as theCRONE (Crontrôle Robuste d'Ordre Non Entier) strategy[26], fractionalPIλDμcontroller [27], and fractional orderlead-lag compensators [28]. To the authors’ best knowl-edge, no attempt has been made in the literature to controlthe retrieval of tethered satellites by the FOC theory.

In this paper, a new fractional order tension control lawis proposed for the retrieval of tethered satellites. Thenonlinear tether dynamics is linearized in the neighbor-hood of a given equilibrium point. By introducing the lineartension feedback and fractional order items, the linearizeddynamic equations become asymptotically stable. Com-pared to the exiting tension control law, the main advan-tages of the proposed fractional order tension control laware: (i) a better control strategy to overcome the instabilityexperienced by the integer order tension control laws; (ii) afaster retrieval than its integer order counterparts, (iii) ahigh efficiency and real-time control law that requires lessfeedback and is easy to implement in engineer applications.

The paper is organized in five sections. Following thisbrief Introduction, Section 2 provides some theoreticalbackground of the dynamics of tethered satellites and

basic concepts for the fractional order calculus. In Section3, the fractional order control law is derived in terms ofstability analysis and model discretization. The simulationresults and discussion are presented in Section 4. Finally,Section 5 concludes the paper.

2. Theoretical background

2.1. Dynamics of tethered satellite system

A tethered satellite system (TSS) usually consists of along and thin tether connecting two satellites at its ends.Generally, the TSS can be modeled in five categories [21]depending on the level of details required in analysis. Tofocus our attention on the control strategy, it is assumedthat the tether is straight and inextensible and its mass isnegligible compared to the main satellite. Due to the largeratio of tether length over satellites’ dimensions, thesatellites are modeled as lumped masses attached to theends of the tether. Moreover, the center of mass of the TSSis assumed to reside in the main satellite throughout thewhole retrieval process. Based upon the assumptions, theTSS motion can be safely decomposed into the orbitalmotion of the main satellite around the Earth and the locallibration motion of tether about the main satellite, asshown in Fig. 1. It is assumed that the TSS orbits the Earthin in a circular orbit and there is no thruster at thesubsatellite. Then, a local orbital reference frame is definedwith the origin at the main satellite. The y-axis of theorbital frame points toward the Earth center. The x-axislies in the orbital plane perpendicular to the y-axis andpointing towards the moving direction of main satellite.The z-axis completes a right-hand coordinate system.

Thus, the equations of motion of the TSS without thrustat the subsatellite can be expressed as per [12]:

€l� l½ _φ2þ cos 2φð _θþΩÞ2�Ω2þ3Ω2 cos 2φ cos 2θ� ¼ �T=m

€θþ2½ð_l=lÞ� _φ tan φ�ð _θþΩÞþ3Ω2 cos θ sin θ¼ 0

€φþ2ð_l=lÞ _φþ cos φ sin φ½ð _θþΩÞ2þ3Ω2 cos 2θ� ¼ 0 ð1Þwhere l is the instant tether length, m is the mass ofsubsatellite, T is the tension in tether,Ω is the orbit rate ofthe TSS, θ and ϕ are the in-plane and out-of-planelibration angles, respectively. The dot denotes the deriva-tive with respect to time.

G. Sun, Z.H. Zhu / Acta Astronautica 104 (2014) 304–312306

Introduce the following dimensionless variables

τ¼Ωt; λ¼ l=L; T̂ ¼ T=½mΩ2L� ð2Þ

Eq. (1) can be rewritten in the dimensionless form,

λ0 �λ½φ02þ cos 2φð1þθ0Þ2�1þ3 cos 2θ cos 2φ� ¼ � T̂

θ00 þ2½ðλ0=λÞ�φ0 tan φ�ð1þθ0Þþ3 cos θ sin θ¼ 0

φ00 þ2ðλ0=λÞφ0 þ cos φ sin φ½ð1þθ0Þ2þ3 cos 2θ� ¼ 0 ð3Þ

where L is the total length of tether before retrieval andthe prime denotes the dimensionless time derivative. T̂ isthe dimensionless tether tension and will be the designedcontrol input in the tension control law. Obviously, this is asingle-input-multiple-output system and the librationangles cannot be controlled directly by the tension.Instead, they are controlled indirectly through the non-linear modal coupling. Eq. (3) shows clearly that the termðλ0=λÞð1þθ0Þ is for the coupling between the tension andin-plane libration while the term ðλ0=λÞφ0 is for thecoupling between the tension and out-of-plane libration.It is noted that the coupling between the tension and out-of-plane libration mode is much weaker (one orderhigher) than the in-plane libration mode.

Redefine the new variablesasx1 ¼ λ; x2 ¼ λ0; x3 ¼ θ; x4 ¼ θ0

; x5 ¼φ; x6 ¼φ0, Eq. (3) istransformed into a set of first order differential equations,such that,

x1 0 ¼ x2

x2 0 ¼ x1x26þx1ð1þx4Þ2 cos 2x5�x1þ3x1 cos 2x5 cos 2x3� T̂

x3 0 ¼ x4

x4 0 ¼ �2 x2=x1� ��x6 tan x5� �ð1þx4Þ�3 cos x3 sin x3

x5 0 ¼ x6

x6 0 ¼ �2 x2=x1� �

x6� cos x5 sin x5 ð1þx4Þ2þ3 cos 2x3h i

ð4Þ

Assume the TSS system is in equilibrium at any pointbetween the completely retrieved and fully deployedstates, such that, x1 ¼ a; x2 ¼ x3 ¼ x4 ¼ x5 ¼ x6 ¼ 0 with0oar1, where a¼1 or 0 means the tethered satellite isfully deployed or completely retrieved. Change the originalof coordinates to the equilibrium point by definingnew coordinates as y1 ¼ x1�a; y2 ¼ x2; y3 ¼ x3; y4 ¼ x4;y5 ¼ x5; y6 ¼ x6. Then, Eq. (4) becomes

y10 ¼ y2

y20 ¼ ðy1þaÞy26þðy1þaÞð1þy4Þ2 cos 2y5

�ðy1þaÞþ3ðy1þaÞ cos 2y5 cos2y3� T̂

y30 ¼ y4

y40 ¼ �2 y2=y1þa

� ��y6 tan y5� �ð1þy4Þ�3 cos y3 sin y3

y50 ¼ y6

y60 ¼ �2 y2=y1þa

� �y6� cos y5 sin y5 ð1þy4Þ2þ3 cos 2y3

h ið5Þ

Thus, the linearized form of Eq. (5) in the neighborhoodof equilibrium (y1 ¼ y2 ¼ y3 ¼ y4 ¼ y5 ¼ y6 ¼ 0) becomes

y10

y20

y30

y40

y50

y60

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼

0 1 0 0 0 03 0 0 2a 0 00 0 0 1 0 00 �2

a �3 0 0 00 0 0 0 0 10 0 0 0 �4 0

2666666664

3777777775

y1y2y3y4y5y6

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

þ

03a� T̂

0000

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;ð6Þ

where the constant 3a is the static tension at theequilibrium.

2.2. Preliminaries of fractional order calculus

The fractional order calculus has many different defini-tions for the fractional order differintegral. Currently, thereare three commonly used definitions in the literature:Riemann–Liouville (R–L), Caputo and Grunwald–Letnikov(G–L). More details about the fractional order calculuscan be found in the Refs. [28–30]. In the current work,we adopt the Caputo definition of the fractional orderderivative, such that,

Dαt f ðtÞ ¼

1Γðn�αÞ

Z t

0

f ðnÞðτÞðt�τÞα�nþ1dτ; n�1oαon ð7Þ

where α is a real number, n is a given integer numberand Γ(�) is the Gamma function defined as ΓðxÞ ¼ R1

0tx�1e� tdt:

It is noted that the fractional order derivative of afunction is related to not only the current state f ðtnÞ butalso the previous states f ðtiÞ; ion. It indicates the frac-tional order derivative possesses memory capacity. Thestability of fractional order calculus is very important to itsapplication in dynamics and control. Matignon pioneeredin this area [31] by checking the condition of fractionalorder system stability by the poles of fractional ordersystem in the complex plane of its pseudo-state space.Subsequently, many stability conditions have been derived[32–34], for instance, the Linear Matrix Inequality (LMI)condition for linear and time-delay systems [32,33], Mit-tag–Leffler stability for nonlinear fractional order systems[34] and so on. Depending on the system characteristics,there are two stability theorems for a fractional ordersystem.

Theorem 1. [31]: A commensurate transfer function F(s)¼T(sα) / R(sα) with the order 0oαo1, where T and R arecoprime polynomials, is Bounded-Input Bounded-Outputstable if and only if for every pAC such that R(p)¼0:

argðpÞ 4απ2

������ ð8Þ

Theorem 1 states that the stable region of a fractionalorder system is generally larger than its integer ordercounterpart, see Fig. 2. The latter can be viewed as aspecial case with the order α¼1.

Theorem 2. [33]: Given a non-commensurate fractionalorder system:

Dα1x1 ¼ c11x1þc12x2þ⋯þc1nxn

Fig. 2. The typical stable and unstable regions of a fractional order system.

G. Sun, Z.H. Zhu / Acta Astronautica 104 (2014) 304–312 307

Dα2x2 ¼ c21x1þc22x2þ⋯þc2nxn

⋮⋮

Dαn xn ¼ cn1x1þcn2x2þ⋯þcnnxn ð9Þ

with all fractional orders αiA(0, 1], i¼0, 1, …, n. Let M be thelowest common multiplier of the denominators ui, i¼0, 1, …,n, where the fractional order αi¼vi /ui, i¼0, 1, …, n, (vi, ui)¼1 and vi, uiAZþ . Set γ¼1/M, then the zero solution of Eq. (9)is global asymptotically stable if all the roots pi of Eq. (10)

pMα1 �c11 �c12 ⋯ �c1n�c21 pMα2 �c22 ⋯ �c2n⋮ ⋮ ⋯ ⋮

�cn1 �cn2 ⋯ pMαn �cnn

����������

����������¼ 0 ð10Þ

satisfies

jargðpiÞj4γπ=2 ð11Þ

3. Control law for tethered satellite retrieval

3.1. Simple tension control law

As a benchmark for the effectiveness of the proposedfractional order tension control law, we first derive asimple tension control law for retrieval as a baseline.Consider the governing equation of the TSS in Eq. (5).The simplest tension control law in the literature has thefollowing form [17]

T̂ ¼ k1y1þk2y2þk3y3þk4y4þk5y5þk6y6þk7 ð12Þ

where the parameters ki; i¼ 1;2;…;7are the control gainsto be determined by the stability condition and controlperformance.

Substituting Eq. (12) into Eq. (6) leads to

y10

y20

y30

y40

y50

y60

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼

0 1 0 0 0 03�k1 �k2 �k3 2a�k4 �k5 �k60 0 0 1 0 00 �2

a �3 0 0 00 0 0 0 0 10 0 0 0 �4 0

2666666664

3777777775

y1y2y3y4y5y6

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

þ

03a�k7� T̂

0000

8>>>>>>>><>>>>>>>>:

9>>>>>>>>=>>>>>>>>;

ð13Þ

For the tether system in Eq. (13) to be stable at theequilibrium, there must be k7¼3a and all eigenvalues ofEq. (13) are located in the open left half-plane. Theeigenfunction of Eq. (13) is

p �1 0 0 0 0k1�3 pþk2 þk3 k4�2a k5 k60 0 p �1 0 00 2

a 3 p 0 00 0 0 0 p �10 0 0 0 4 p

���������������

���������������¼ p6þk2p5þð8�2k4=aþk1Þp4þð7k2�2k3=aÞp3

þð7�8k4=aþ7k1Þp2þð12k2�8k3=aÞp�36þ12k1¼ 0 ð14Þ

It is interesting to note that the parameters k5 and k6 arenot presented in the eigenfunction (14). This implies thatthe stability is independent to the feedback gain k5 and k6.In other words, the simple tension control law is comple-tely decoupled from the out-of-plane mode in the linear-ization process due to the high-order nonlinear couplingnature of the out-of-plane angle φ with the other twovariables λ;θ is eliminated. Thus, an out-of-plane librationcannot be controlled effectively alone by the tensionfeedback. Previous work showed that a thrust was neces-sary to suppress the libration angle φ to zero quickly inRef. [17]. It should be noted that this limitation is due tothe simple tension control strategy. Any control law basedon this control strategy will suffer the same limitation.

The stability condition of Eq. (13) can be expressed asper Routh–Hurwitz criterion

k240; 1�2ak4þk1þ2k3

ak240; 3k1�940;

12k2�6ak2k4þ

4ak3�

4k3k4a2

�2k3k1a

� 4k23a2k2

40 ð15Þ

From Eq. (15), the stable ranges of (k1, k2) can be derived ask143 and k240, explicitly. The stable range of k3 can bedetermined as k3¼0 by the KTC asymptotical stabilitytheorem [17]. Substituting k3¼0 into Eq. (15) leads to

k4oa21þk1ð Þ ð16Þ

Obviously, the k4 will approaches to zero as the tether isretrieved, i.e., a-0. For the sake of simplicity in construct-ing the tension control law, it is assumed k4¼0 andk5¼k6¼0 in the following work. Then, the simple tensioncontrol law for the tether retrieval becomes,

T̂ ¼ k1y1þk2y2þ3a; with k14 3; k2 4 0; 0oar1

ð17ÞBy varying the parameter a from 1 to 0, the tether canbe retrieved accordingly. It should be noted that thesimple tension control law cannot be used at a¼0 since

-0.25-0.2-0.15-0.1-0.0500.050.1

-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Initial Position

X

Y

Initial Position

X

Y

Fig. 5. Trajectory of subsatellite in the orbital plane in retrieval process.

G. Sun, Z.H. Zhu / Acta Astronautica 104 (2014) 304–312308

the linearized system Eq. (6) becomes singular. Further-more, the simple tension control law is asymptoticallystable at the equilibrium point a due to the application ofthe KTC theorem and no limitation being imposed on themagnitude of libration angle as the system approaching tothe equilibrium. As a result, the tether may wind aroundthe main satellite when a-0, which will be a catastrophicfailure. The performance of the control law is illustratednumerically as follows.

Assume that the TSS is running in a circular orbit with20 km long tether at an altitude of 220 km. The corre-sponding orbital rate is 1.1804�10�3 rad/s. To circumventthe singularity at a¼0, a small value (a¼0.02) is usedinstead in the analysis. The initial condition for retrieval isy1¼1, y2¼0, y3¼�0.02, y4¼0, y5¼0, y6¼0.02, where theconditions y3¼�0.02 and y5¼0.02 are randomly selectedto introduce an initial perturbation in libration angles.Furthermore, the control gains (k1, k2) are optimized forthe tethered satellite to be retrieved within two orbits,with k1¼4 and k2¼2.8. The system responses are shownin Figs. 3–6. Fig. 3 shows that the subsatellite can beretrieved by the simple tension control law within threeorbits. However, the in-plane libration angle shown inFig. 4 increases as the subsatellite approaches to the main

Fig. 3. Time history of dimensionless teth

'

Fig. 4. Time history of in-plane libration angle and velocity of retrieval. Note th

satellite. After one and half orbits, the tether length hasbeen reduced to less than 5% of the total length and thelibration angle suddenly increases over 1801, which implies

er length and velocity in retrieval.

at the unit for the libration velocity is degree per dimensionless time (τ).

'

Fig. 6. Time history of out-of-plane libration angle and velocity of retrieval. Note that the unit for the libration velocity is degree per dimensionless time (τ).

G. Sun, Z.H. Zhu / Acta Astronautica 104 (2014) 304–312 309

the tether wound around the main satellite or the sub-satellite hit the main satellite. This can be best illustrated bythe trajectory of the subsatellite in the orbital plane in Fig. 5,where the subsatellite starts to wind around the mainsatellite when the tether length is very short. Moreover,Fig. 6 shows that there is violent out-of-plane librationmotion for the non-zero initial value y5 ¼ 0:02. Such theresults are expected as discussed before.

3.2. New fractional order tension control law

To retrieve the subsatellite stably and quickly, theexisting control strategies commonly employ two stagecontrol laws: fast retrieval followed by slow retrieval afterthe tether is less than 1% of its original length [35]. In thecurrent work, a new fractional order tension control isproposed, which is able to control the system performancewith the information of previous states. Based on thesimple tension control law outlined above, the fractionalorder tension control law is defined as follows,

T̂ ¼Dαy2�y2'þk1y1þk2y2þ3a ð18Þwhere Dαy2 is the Caputo definition of fractional orderdifferential. Substituting Eq. (18) into (6) leads to a non-commensurate fractional order system in the neighbor-hood of the equilibrium point, such that,

y10

Dαy2y3

0

y40

y50

y60

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼

0 1 0 0 0 03�k1 �k2 0 0:04 0 00 0 0 1 0 00 �100 �3 0 0 00 0 0 0 0 10 0 0 0 �4 0

2666666664

3777777775

y1y2y3y4y5y6

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;ð19Þ

Here, it is assumed a¼0.02 in order to avoid the singular-ity at a¼0. According to the Theorem 2, Eq. (19) canasymptotically approach to the equilibrium point for anygiven fractional order α with properly selected values of(k1,k2). Furthermore, Theorem 1 indicates that the stable

region of a fractional order system is always larger thanits integer order counterpart. Thus, we choose the samevalues for (k1,k2) as in the integer order control law inEq. (17) to satisfy the stability condition instead of findingthe eigenvalues of the fractional order system in Eq. (19).By replacing the fractional order derivative with thefractional order integral, the fractional order tension lawis written as

T̂ ¼Dαy2�y20 þ4y1þ2:8y2þ0:06¼ y2

0ðI1�αy2�1Þþ4y1þ2:8y2þ0:06 ð20Þ

and

y10

Dαy2y3

0

y40

y50

y60

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

¼

0 1 0 0 0 03 0 0 0:04 0 00 0 0 1 0 00 �100 �3 0 0 00 0 0 0 0 10 0 0 0 �4 0

2666666664

3777777775

y1y2y3y4y5y6

8>>>>>>>>><>>>>>>>>>:

9>>>>>>>>>=>>>>>>>>>;

ð21Þwhere I1�αdenotes the fractional order integral operatorsatisfying D1�αðI1�αy2Þ ¼ y2.

Next, we will develop a numerical procedure to imple-ment the fraction order controller. Assume the fractionalorder α¼0.05. The fraction order integral I1�α in Eq. (21)can be approximated by the Laplace transform and proce-dures in [36], such that,

LfI0:95y2g ¼1

s0:95¼ 2:977� 10�7s2þ0:6159sþ78:481

1:1289� 10�7s3þ0:3793s2þ78:481sþ1ð22Þ

The inclusion of the feedback of derivative y2’ (accel-eration) in Eq. (20) may lead to some unwanted oscilla-tions in the system responses. To avoid the possibleoscillations, the Laplace transform is modified by followingitem,

s’s

T1sþ1ð23Þ

Fig. 7. Comparison of time history of dimensionless tether length with different control laws for retrieval.

Fig. 8. Comparison of time history of in-plane libration angle with different control laws.

Fig. 9. Comparison of time history of out-of-plane libration angle with different control laws.

G. Sun, Z.H. Zhu / Acta Astronautica 104 (2014) 304–312310

G. Sun, Z.H. Zhu / Acta Astronautica 104 (2014) 304–312 311

where T1 is the parameter to be determined by controlrequirement. In the current work, it is defined as T1¼0.2.

Assume the sampling time step is Ts¼0.002, Eq. (22) is

discretized by the Tustin operator [37], s¼ 2Ts

1� z� 1

1þ z� 1

h i, such

that,

Z�1 1s0:95

� �� 0:00151z3þ0:00185z2�0:000832z�0:001173

z3�0:6577z2�0:999zþ0:6567

ð24Þ

Hence, the implementation procedure of the fractionalorder tension control law in Eq. (20) is established.

Using the same initial condition employed in the caseof the integer order tension control law, the performanceof the newly devised fractional order control law is shownin Figs. 7–11 in comparison with its integer order counter-part. The solid lines denote the fractional order controllerwhile the dotted lines represent the simple integer ordertension controller. The results show that the fractionalorder control provides a more stable and fast retrieval thanits integral order counterpart. The fractional order control

X

Y

Fig. 10. Comparison of subsatellite trajectory in orbital plane withdifferent control laws.

0

0.05

0.1

0.15

0.2

T̂

0

0.05

0.1

0.15

0.2

T̂T̂

Fig. 11. Comparison of time history of tethe

works well even in the vicinity of singularity at a¼0.01.The tether length reduction is smoother with the fractionalorder controller as shown in Fig. 7. Fig. 8 shows themaximum in-plane libration angle is less than 171. Fig. 9illustrates that, since the existence of the nonlinear cou-pling in the governing Eq. (5), the maximal value of out-of-plane libration (371) is also smaller than that of the integerorder tension control (421), and approaches to zero morequickly. Fig. 10 indicates the subsatellite is retrieved with-out the tether winding or hitting the main satellite whenthe tether length is short under the fractional ordercontrol. Finally, Fig. 11 shows that the tether tension isalso bounded in an acceptable region even though thereare some oscillations in the initial stage of retrievalcompared with the integer order tension controller.

4. Conclusion

This paper investigates the stability control problem ofthe retrieval of a tethered satellite system, which isintrinsically unstable. The originality of the current workis to control the retrieval of a tethered satellite stably andquickly by a new fractional order tension control law. Thegoverning equation of tethered satellite systems andclassic linear feedback tension control law were firstlyreviewed and examined as a benchmark test. The newlyproposed fractional order control law is developed from itsinteger order counterpart to ensure the stability of thenew control law conservatively. The new controller hasbeen discretized and implemented by the Laplace trans-form and Tustin operator for practical application. Unlikewith the classic integer order control law, which is basedon the feedback of previous state and memoryless, thefractional order control law has the memory of previousstates and thus controls the tether retrieval more smoothlywhile maintain the retrieving speed. It is noted that thesimple linear tension control strategy is not effective incontrolling the out-of-plane libration because the feedbackof the states of that mode does not affect the stability ofthe tethered system. As expected, the linear fractionalorder tension control law, which is based on the same

r tension with different control laws.

G. Sun, Z.H. Zhu / Acta Astronautica 104 (2014) 304–312312

control strategy, is not effective in controlling the out-of-plane libration neither. The effectiveness and advantage ofthe new fractional order tension control law for in-planmotion control is demonstrated numerically by compari-sons with its integer order counterpart. It not only retrievesthe subsatellite without winding around the main satellite,but also provides a better control performance with smallerin-plane libration angles.

Acknowledgment

This work is supported by the Discovery Grant and theDiscovery Accelerator Supplement Grant of the NaturalSciences and Engineering Research Council of Canada andpartially supported by the National Natural Science Foun-dation of China (No. 61104112).

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