Robust Attitude Coordination Control of Multiple-teamSpacecraft Formation under control input saturation

Xiangdong Liu1, Yaohua Guo1, Pingli Lu1

1. School of Automation, Beijing Institute of Technology, Beijing 100081, P. R. ChinaE-mail: pinglilu@bit.edu.cn

Abstract: This paper presents a robust attitude coordination methodology for multiple-team spacecraft formation(MTSF) inpresence of control input saturation. The main idea is to construct a novel sliding mode surface with a tunable vector havingthe following two properties: 1) the gradient of the tunable vector is related to saturation constraints; 2) the tunable vector hasconsensus-like property related to the formation’s communication topology. The robustness against matched perturbations andthe stability of the formation system are proven by Lyapunov theory. Moreover, we give extensions of our control method tothe case with multiple communication delays by solving linear matrix inequalities(LMIs) to choose control parameters properly.Finally, simulation results are presented for attitude coordination for 3-teams spacecraft formation.

Key Words: Attitude Coordination, Multiple-team Spacecraft Formation, Control Input Saturation, LMIs

1 INTRODUCTION

Spacecraft formation flying(SFF) is a promising and at-

tractive enabling concept for lots of scientific and military

missions. Some examples of potential applications of SF-

F include monitoring of the earth and its surrounding atmo-

sphere, deep space imaging and exploration, military surveil-

lance instruments, and so on. Attitude coordination control

is one of the most challenging techniques for SFF and has re-

ceived wide attention[1]–[7]. Various solutions for attitude

coordination in literatures can be broadly categorized into

three main approaches, namely, the leader-follower[1], the

virtual structure[2, 3] and the behavior based approach[4]–

[7].

In the leader-follower(LF) approach, [1] uses the state-

dependent Riccati equation method of nonlinear regulation

to control the attitudes of a LF satellite pair. Ren and

Randal[2] point out the advantages and limits of LF ap-

proach, i.e., easy to understand and implement, while being a

single point of failure. They present formation control ideas

for multiple spacecraft using virtual structure(VS)[2], and

more importantly, introduce formation feedback from space-

craft to the VS. Later, [3] combines the strength of decentral-

ized control and VS approach to improve the VS formation

control scheme. The behavioral approach is a decentralized

strategy, and has the advantages such as flexibility, reliabil-

ity, and robustness. In [4], two control strategies(velocity

feedback and passivity-based damping) based on emergen-

t behavior approaches were presented for maintaining atti-

tude alignment among a group of spacecrafts under a bidi-

rectional ring topology. A class of decentralized coordinated

attitude control laws using behavior-based control is devel-

oped in [5], in which the choice of behavior weights defines

the coordination architecture. Ren[6] analyzes the key role

of interspacecraft information exchange in attitude align-

ment and extends some consensus algorithms from single-

or double-integrator dynamics. Unlike the literatures men-

tioned previously, [7, 8] and [9] addressed the robust control

for attitude synchronization against model uncertainties and

external disturbances under directed communication topol-

ogy. The decentralized adaptive sliding-mode control law

proposed in [7] ensures that each spacecraft attains desired

attitude while maintaining attitude synchronization with oth-

er spacecrafts in the formation. Furthermore, [8] and [9]

present sufficient conditions for finite time stability of at-

titude synchronization in presence of external disturbances.

Unfortunately, attitude alignment of all spacecrafts can’t per-

form some particular complicated space missions, to name a

few, the 25-meter aperture virtual structure gossamer tele-

scope in GEO[10], the solar power satellite system[11], and

the Canadian team’s GEO satellite cluster[12]. Therefore

multi-team spacecraft formation(MTSF) attitude coordina-

tion, which has multiple desired attitude trajectories, is an

important point to pay much attention.

For single spacecraft, control input saturation is an signif-

icant problem encountered in practice and affects the perfor-

mance of attitude tracking. Nevertheless, in current litera-

tures, the attitude coordination control for SFF under control

input saturation has not received much attention. Tandale

etc.[13] present an adaptive control method to modify the

reference attitude trajectory on saturation based on a nov-

el ‘hedging signal’. Sean [14] develops an adaptive scheme

too, but use it to tune the control parameters. In case of

attitude coordination of spacecraft formation, control input

saturation becomes an even more important issue to be dealt

with. Most existing literatures addressing attitude coordina-

tion assume that the control authorities of each spacecraft is

large enough. However, when some spacecrafts are saturat-

ed, while others not, the spacecrafts under control input sat-

uration will fall behind and thus degrade the performance of

attitude coordination. Another important problem encoun-

tered in practice is that of communication time delays, and

only few papers dealing with this problem due to the nonlin-

earity of the attitude dynamics[15].

The present paper focuses on the development of an dis-

tributed robust sliding mode controller for attitude coordina-

tion of MTSF under control input saturation. Adding a nov-

el tunable vector into typical sliding mode(SM) vector, the

states of MTSF system can keep in the new SM surface even

under control input saturation. Further, extension of our pro-

posed control law into communication delay case is achieved

by solving LMIs to choose control parameters properly.

The remainder of this paper is organized as follows. Atti-

Proceedings of the 33rd Chinese Control ConferenceJuly 28-30, 2014, Nanjing, China

1389

tude dynamics, graph theory and control objective are briefly

described in the next section. Section 3 shows our main

contribution including the design of tunable vector and the

extension. Some simulations are provided in section 4 to

illustrate the effectiveness of our proposed control method.

Finally, section 5 concludes the paper.

The notations used throughout this paper is as follows.

The notation Rn×n represents the n × n dimensional Eu-

clidean space and In denotes the identity matrix of size n.

Let diag(r1, r2, · · · , rn) denotes a diagonal matrix with el-

ements ri, i = 1, 2, · · · , n on its diagonal. For any complex

matrix A, we denote its transpose by AT and A > 0 means

A is a positive definite matrix. ⊗ stands for Kronecker prod-

uct and ‖ · ‖ denotes the standard Euclidean vector norm or

induced matrix norm, as appropriate.

2 PRELIMINARIES

2.1 Spacecraft Attitude Kinematics and DynamicsConsider a group of n spacecrafts modeled as rigid bodies,

the equations of the ith spacecraft are given by

Jiωi = −ω×i Jiωi + τi + zi (1)

qi = M (qi)ωi, i = 1, 2, · · · , n (2)

where Ji = Ji + δJi ∈ R3×3 is the actual inertia matrix of

the ith spacecraft, with Ji and δJi denote the nominal part

and uncertain part, respectively; τi ∈ R3 is the vector of con-

trol torque of the ith spacecraft commanded by the controller

and zi is the bounded external disturbances; ωi ∈ R3 and

qi ∈ R3 are, respectively, the inertial angular velocity and

modified Rodrigues parameters(MRPs) of the ith spacecraft.

The using of MRPs to describe attitude has an advantage of

being valid for eigenaxis rotations up to 360 deg(more de-

tails about MRPs can be found in [16]). Further, the notation

v× for a vector v = [v1, v2, v3]T is used to denote the

skew-symmetric matrix

v× =

⎡⎣ 0 −v3 v2

v3 0 −v1−v2 v1 0

⎤⎦

The Jacobian matrix M(qi) ∈ R3×3 for the MRPs is given

by

M(qi) =1

2

(1− qTi qi

2I3 + q×i + qiq

Ti

)(3)

where I3 denotes the identity matrix. From now on, Mi is

used to represent M(qi) for simpleness.

2.2 Graph theroylet G = (V , E ,A) be a weighted directed graph describ-

ing the communication topology of the spacecraft forma-

tion, which consists of a node set V = {1, 2, · · · , n}, an

edge set E ⊆ V × V , and a weighted adjacency matrix

A = [aij ] ∈ Rn×n with nonnegative adjacency elements

aij . An edge (i, j) ∈ E in a weighted directed graph indi-

cates the jth spacecraft can receive information from the ithspacecraft and i is called parent node while j the child node.

It is assumed that the graph has no self-loops, meaning that

(i, j) ∈ E implies i �= j. The weighed adjacency matrix Ais defined such that aij > 0 if and only if (i, j) ∈ E and

aij = 0 otherwise. Moreover, we assume aii = 0 for all

i ∈ V . The set of neighbors of spacecraft i is denoted by

Ni = {j : (i, j) ∈ V}. The Laplacian L ∈ Rn×n of graph

G is defined element-wise as

Lij =

⎧⎨⎩

−aij if i �= j;N∑

k=1,k �=k

aik if i = j.

Obviously, all the eigenvalues of the graph Laplacian Lhave a nonnegative part(follows from Gershgorin’s Theo-

rem), which will be used to drive stability proof for the pro-

posed control laws.

2.3 Problem StatementTo achieve attitude coordination control of MTSF, space-

crafts in formation necessarily exchange some of their states

information to achieve and maintain a given formation, and

simultaneously track their reference trajectory. The follow-

ing assumptions are then considered to facilitate our solving

of attitude coordination problem.

Assumption 1 The inertia matrix uncertainty δJi is as-sumed to satisfy ‖δJi‖ ≤ δm for i = 1, 2, · · · , n, whereδm is a positive constant.

Assumption 2 The disturbance zi, i = 1, 2, · · · , n isbounded by a positive constant zm i.e., ‖zi‖ ≤ zm.

Assumption 3 The spacecraft formation is divided intomultiple teams by the nonidentical reference attitude signalsqdi , i = 1, 2, · · · , n. And we make assumptions that the ref-erence attitude qdi and its first two derivatives(i.e.,qdi and qdi )are bounded.

With the above assumptions, our control objective is to de-

sign control laws for each spacecraft with control input sat-

uration constraints, under Assumptions 1,2,and 3, such that

the following problems are solved:

Problem i: Attitude coordination control of MTSF under

a general directed communication topology and in face of

matched perturbations.

Problem ii: Attitude coordination control of MTSF in p-

resence of multiple time-delays among spacecrafts.

3 MAIN RESULTS

In this section, based on the existing results on sliding

mode control, we will first present the design procedure of

attitude coordination control law for MTSF with no con-

straints on control input. Then a tunable vector is introduced

to handle the actuator saturation limits. We further extends

the proposed control method to the case with multiple com-

munication delays among spacecrafts.

3.1 Equivalent Control-Based Sliding Mode ControllerThe multi-spacecraft sliding vector proposed in this sub-

section is similar to the one introduced in [17].

S =[sT1 , s

T2 , · · · , sTn

]T(4)

In (4), si ∈ R3, i = 1, 2, · · · , n is given by

si = Miωi + riqi + ki

n∑j=1,j �=i

aij(qi − qj) (5)

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with ωi and qi are the tracking errors defined by

ωi = ωi −Riωdi , i = 1, 2, · · · , n;

qi =qdi

(|qi|2 − 1)+ qi

(1− |qdi |2

)+ 2q×i q

di

1 + |qi|2|qdi |2where Ri is the rotation matrix from the ith spacecraft’s de-

sired orientation to its body-fixed frame, ri > 0 and ki > 0are the control weight parameters determining the trade off

between station-keeping and formation-keeping behavior.

For developing control law for each spacecraft, differenti-

ating (5) with respect to time t, we obtain

si = ψi + ρi +MiJ−1i τi (6)

where

ψi = Mi

(−J−1i ω×

i Jiωi + ω×i Riω

di −Riω

di

)+

ri ˙qi +

n∑j=1,j �=i

aij( ˙qi − ˙qj) (7)

ρi = MiJ−1i Δi (8)

In (8), Δi � zi − ω×i δJiωi − δJi( ˙ωi + Riω

di + ω×

i ωi)represents the influences caused by bounded matched per-

turbations(i.e., inertia matrix uncertainty and external distur-

bances). Due to the bounds of perturbations and reference

attitude signals, it’s reasonable to make another assumption

as given below.

Assumption 4 There exists a nonnegative scaler function ϑi

such that

‖Δi‖ ≤ ϑi(ωi, δJi, zi), i = 1, 2, · · · , n (9)

The control input of ith spacecraft is then chosen in the

form

τi = τeqi + τ roi , i = 1, 2, · · · , n (10)

where τeqi denotes the equivalent control component that

keeps si = 0 for all time and directs the system states toward

the desired trajectory; τ roi is the variable structure compo-

nent, which ensures that the sliding surface si = 0 is attrac-

tive and can reach sliding surface in finite time[18]. Con-

sidering (6) and Assumption 4, τeqi and τ roi can be obtained

as

τeqi = −JiM−1i ψi (11)

τ roi = − [ϑi(ωi, δJi, zi) + ηi] Ξ(si) (12)

where ηi is a positive constant, Ξ(si) is defined by Ξ(si) �[sgn(s1i ) sgn(s

2i ) sgn(s

3i )]

T , with sji , j = 1, 2, 3 represents

the jth element of si and sgn(·) denotes the sign function.

The stability of the MTSF system and the reaching of s-

liding surface can be easily shown by choosing a standard

Lyapunov candidate as V = STS. Here we only need to

give explanations of the response when S = 0. Using the

Kronecker product and considering (5), the multi-spacecraft

sliding vector in (4) can be written as

S = ˙Q+ ΛQ (13)

where the terms Q = [qT1 , qT2 , · · · , qTn ]T ∈ R

3n and

Λ = R ⊗ I3 + (K ⊗ I3)(L ⊗ I3), in which R =diag(r1, r2, · · · , rn) ∈ R

n×n , K = diag(k1, k2, · · · , kn) ∈R

n×n and L is the Laplacian matrix of G. Since all the

eigenvalues of the graph Laplacian L have a nonnegative

part, together with R and K are positive definite matrixes,

S = 0 leads to the fact that

limt→∞ ‖Q‖ = 0, lim

t→∞ ‖ ˙Q‖ = 0 (14)

Obviously, (14) implies that

limt→∞ ‖ωi‖ = lim

t→∞ ‖qi‖ = 0, i = 1, 2, · · · , n (15)

Hence, it can be concluded that on the multi-spacecraft sur-

face S = 0, the attitude and angular velocity of each space-

craft converge to the desired trajectory and become unsensi-

tive to the matched perturbations.

Remark 1 One of the problems the preceding control lawrelates to is that of input chattering, which is commonlysolved by using either the saturation function or some ap-proximate sign functions to replace sign function. One ex-ample of these approximate sign functions is of the formf(x) = x/(|x|+ ε), where ε > 0 is constant and commonlychosen to be sufficiently small.

Remark 2 Another significant problem the preceding con-trol law faces is not taking account of the control input sat-uration. In (6), ψi depends on the instantaneous responseof the system and ρi mainly depends on the matched pertur-bations. To handle the actuator saturation constraint, onefeasible way is to modify the response of the system. This isthe main motivation of the introducing of a tunable vector innext subsection.

3.2 Sliding Mode Controller Under Actuator Satura-tion Constraints

For MTSF, a tunable vector needs to be carefully designed

such that it is characterized by consensus property, which

handles the saturation of MTSF as a whole. We design the

tunable vector Ti ∈ R3, i = 1, 2, · · · , n, whose differential

equation is

Ti = −Πi + θi(t), (16)

Πi = riTi + ki

n∑j=1,j �=i

aij(Ti − Tj)

where ri and ki are defined in (5), which specify how fast

Ti converges to zero, θi is a tunable function that modifies

the response of the system, which will be discussed in detail

later.

Adding the tunable vector into the sliding vector in (5), a

new sliding mode surface is obtained in the form

si = si + Ti, i = 1, 2, · · · , n (17)

Then by utilizing the proposed control method in previous

subsection, we get

τ cali = τeqi + τ roi , i = 1, 2, · · · , n (18)

where τeqi and τ roi are denoted by

τeqi = −JiM−1i (ψi −Πi + θi) (19)

1391

τ roi = − [ϑi(ωi, δJi, zi) + ηi] Ξ(si) (20)

The calculated control is obtained from (18), which can

be limited into a bounded set Ψui = {u : −umi ≤ ‖u‖ ≤umi, u ∈ R

3 i = 1, 2, · · · , n} by tuning θi, where umi ∈ R

denotes the bound of input torque of the ith spacecraft. Be-

cause θi acts as a disturbance into the dynamics of tunable

vector described by (16), it is desirable that θi = 0 when the

controls becomes unsaturated. For this purpose, we intro-

duce a virtual applied control input given by

τvi = −JiM−1i (ψi −Πi) + τ roi , i = 1, 2, · · · , n (21)

Consider the actuator saturation limits, the controller that is

practically applied is

τappi =

{τvi if‖τvi ‖ ∈ Ψui ;uimsgn(τvi ) else.

(22)

Then we choose θi as

θi = MiJ−1i (τappi − τvi ) , i = 1, 2, · · · , n (23)

Taking (18) together with (23) into account, we obtain that

τ cali = τappi all the time and sliding vectors in (17) act as if

the control authorities for each spacecraft are always suffi-

ciently large. Hence, the actual control input (22) can ensure

that the MTSF’s states reach and keep in the sliding surface

(17). Further, (23) shows that when the virtual control input

is not saturated θi = 0, which is the base of analysis in the

sequel.

A natural question arises: Does the tunable vector Ti, i =1, 2, · · · , n converge to zero to realize attitude coordination

and tracking. Writing (16) into matrix form, we have

T = −ΛT +Θ (24)

where T = [TT1 , TT

2 , · · · , TTn ]T , Θ = [θT1 , θ

T2 , · · · , θTn ]T ,

and Λ is defined the same as (13). In (24), Θ is related to

control input saturation and −ΛT is the stabilizing compo-

nent. For any feasible reference attitude signals, the control

should be unsaturated for some period of time and in such

cases Θ = 0. Hence, we can conclude that when the control

goes out of saturation, T converges exponentially to zero.

Remark 3 For global robustness against matched pertur-bations, one should choose the initial values of Ti, i =1, 2, · · · , n such that si = 0 from the very beginning. Itneeds to note that the calculating of si and θi is distributiveand the information can be obtained by communication linksbetween spacecrafts.

Remark 4 The effectiveness of the tunable vector is basedon modifying the dynamic response of MTSF through θi de-fined by (23) when the control is saturated. And after theactuator goes out of saturation, θi = 0 and sliding surface(17) becomes a conventional time varying one.

3.3 Extension to Communication Delays CaseThis subsection will extend our proposed control law to

the case where communication delays can not be ignored.

We assume that communication delays are constant and not

identical among spacecrafts. A new multi-spacecraft sliding

mode surface is constructed, which is in the form

S = [sT1 , sT2 , · · · , sTn ]T (25)

where si, i = 1, 2, · · · , n is defined by

si = Miωi +Υiqi −n∑

j=1,j �=i

aij qj(t− dij) + Ti (26)

Ti has the same differential dynamic as (16), dij is the time

delay when information flows from jth to ith spacecraft, and

Υi ∈ R3×3 is the parameter needed to be designed. Before

proceeding our control design, we first introduce a lemman

proven by [19].

Lemma 1 Consider the linear time-delay system describedby

x(t) = A0x(t) +N∑i=1

Aix(t− di) (27)

where x(t) ∈ Rn denotes the state vector, di > 0 are the

delay durations and Ai ∈ Rn×n, i = 0, 1, · · · , N , are con-

stant real matrices. The system is stable if⎡⎢⎢⎢⎢⎢⎣

AT0 P + PA0 +

N∑i=1

Qi PA1 · · · PAN

AT1 P −Q1 0...

. . .AT

NP 0 −QN

⎤⎥⎥⎥⎥⎥⎦ < 0

(28)

where P and Qi ∈ Rn×n, i = 1, 2, · · · , N are symmetric

positive definite matrices.

With respect to the system (25), when the multi-spacecraft

sliding mode surface S = 0 and Ti = 0, the dynamic of the

tracking errors can be described in the form of Lemma 1.

More precisely

˙Q = −ΥQ+n∑

i=1,j=1

AijQ(t− dij) (29)

where Q is the same with the form in (13), Υ =diag(Υ1,Υ2, · · · ,Υn) ∈ R

3n×3n, and Aij = Aij ⊗ I3,

with Aij ∈ Rn×n is defined such that all the elements of it

is 0, except for the ijth equalling to aij .

Then using Lemma 1, the MTSF system is stable and thus

the control problem ii is solved if the following inequality is

satisfied.⎡⎢⎢⎢⎢⎢⎣

KT +K +n2∑i=1

Q′i P ′A11 · · · P ′Ann

AT11P

′ −Q′11 0

.... . .

ATnnP

′ 0 −Q′nn

⎤⎥⎥⎥⎥⎥⎦ < 0

(30)

where P ′, K, and Q′ij , i, j = 1, 2, · · · , n are all positive

definite matrices and dimensions are 3n × 3n. Further,

Υi, i = 1, 2, · · · , n can be calculated by P ′−1K.

Remark 5 Time-delay independent condition for the stabili-ty of MTSF system is established in terms of LMIs (30), read-ily solvable by available numerical software. In fact, an ap-propriate Υi needs to be sufficiently large to satisfy (30). Ancomprehensive essence behind this fact is that time delayscan be neglected when Υi is much larger than Aij .

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Table 1: Simulation parameters

Sp Inertial matrix Initial MRPs Initial ωi

1 diag(180,150,90) [-0.1,0.1,-0.3]T [0.03,0.02,0.05]T

2 diag(90,180,180) [ 0.1,0.1,-0.5]T [0.05,0.06,0.00]T

3 diag(160,160,130) [-0.2,0.3,-0.6]T [0.02,0.02,0.02]T

4 diag(80,150,110) [0.0,0.0,-0.3]T [0.01,0.03,0.04]T

5 diag(100,180,150) [0.3,0.2,-0.1]T [0.02,0.01,0.02]T

6 diag(120,180,160) [0.0,-0.1,0.1]T [0.03,0.02,0.05]T

7 diag(130,160,180) [-0.2,0.1,0.0]T [0.02,0.05,0.03]T

Fig. 1: Seven satellites communication topology

4 SIMULATIONS

Simulation results are presented in this section to illustrate

the performance of the proposed control law. A formation

with seven spacecrafts divided into three groups is consid-

ered for simulation. The nominal inertial matrices, initial

attitudes and angular velocities of MTSF are shown in Table

1. The communication topology of the MTSF is shown in

Fig. 1.

To validate the robustness of the proposed control law

(17), the actual inertial matrices is assumed to be time-

varying and the error bound with respect to the nominal iner-

tial matrices does not exceed 10%. Identical sinusoidal-wave

disturbances as in (31) are introduced to each spacecraft.

di = 0.1[2 sin(0.05πt), sin(0.02πt), 2.5 sin(0.04πt)]T

(31)

Attitudes coordination of each team are shown in Fig.2.

And Fig.3 shows that the actual control inputs of MTSF are

saturated from the beginning, and after 16 seconds all of the

control inputs go out of saturation. We can see that each team

of spacecrafts synchronize their attitudes 20 seconds later

and obtain their desired attitude trajectory before 40 seconds,

which implies the robustness against matched perturbations

even under control inputs saturation.

Fig. 4 and 5 demonstrate the principle and effectiveness of

the tunable vector introduced by (16). The dynamic respons-

es of the tunable vectors Ti, i = 1, 2, · · · , n are shown in

Fig.4. The initial values of Ti are chosen such that si(0) = 0in (17). It needs to be noted that the tunable vectors in Fig.

4 don’t converge to zero exponentially due to control input

saturation. Fig.5 shows the effectiveness of the tunable vec-

tor, i.e., sliding mode surfaces in (17) almost stay in zero all

the time even under control input saturation.

0 20 40 60 800.2

0.1

0

0.1

0.2

0.3

0.4

Time/s

σ1

satellite1,2,3

satellite6,7

satellite4,5

0 20 40 60 800.1

0.05

0

0.05

0.1

0.15

0.2

0.25

0.3

Time/s

σ2satellite4,5

satellite6,7

satellite1,2,3

0 20 40 60 800.6

0.5

0.4

0.3

0.2

0.1

0

0.1

0.2

Time/s

σ3

satellite1,2,3

satellite4,5

satellite6,7

Fig. 2: Attitude coordination of MTSF under saturation

0 20 40 60 805

0

5

T1 ai(N

.m)

0 20 40 60 805

0

5

T2 ai(N

.m)

0 20 40 60 805

0

5

Time/s

T3 ai(N

.m)

i=1,2,3 i=4,5 i=6,7

Fig. 3: Actual control inputs of multi-team spacecraft for-

mation

1393

0 20 40 60 800.1

00.1

T(1)

ii=1,2,3 i=4,5 i=6,7

0 20 40 60 800.1

00.1

T(2)

i

0 20 40 60 800.1

00.1

Time/s

T(3)

i

Fig. 4: Dynamic responses of tunable vector in (16)

0 20 40 60 80101

x 10 3

S1 i

0 20 40 60 80101

x 10 3

S2 i

0 20 40 60 80101

x 10 3

Time/s

S3 i

i=1,2,3 i=4,5 i=6,7

Fig. 5: Sliding mode surface of (17)

5 CONCLUSIONS

This paper derived and validated a sliding mode method

based robust attitude coordination control for MTSF under

control input saturation, which is achieved by introducing a

tunable vector added into the sliding mode surface.The tun-

able vector handles the saturation of MTSF as a whole, and is

related to calculated control inputs and bounds of control au-

thorities. The extension to case with communication delays

is based on choosing proper control parameters by solving

LMIs. Simulation results shows the response of tunable vec-

tor and its effectiveness on handling control input saturation.

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