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Received August 27, 2019, accepted September 23, 2019, date of publication October 11, 2019, date of current version October 28, 2019.
Digital Object Identifier 10.1109/ACCESS.2019.2946961
Fuzzy Disturbance Observer-Based Sliding Mode
Control for Liquid-Filled Spacecraft With Flexible
Structure Under Control Saturation
LIQIAN DOU 1, MIAOMIAO DU1, XIUYUN ZHANG 1, HUI DU1, AND WENJING LIU2
1School of Electrical and Information Engineering, Tianjin University, Tianjin 300072, China
2Science and Technology on Space Intelligent Control Laboratory, Beijing Institute of Control Engineering, Beijing 100101, China
Corresponding author: Xiuyun Zhang (zxy_11@tju.edu.cn)
This work was supported in part by the Natural Science Foundation of China (NSFC) under Grant 61573060, Grant 61873340,
Grant 61673294, and Grant 61773279, and in part by the Joint Science Foundation of Ministry of Education of China under
Grant 6141A0202304.
ABSTRACT This paper investigates a fuzzy disturbance observer (FDO)-based terminal sliding mode
control (TSMC) strategy for the liquid-filled spacecraft with flexible structure(LFS-FS) under control
saturation. Firstly, a novel FDO is designed to estimate the lumped uncertainty, including the inertia
uncertainty, external disturbance, the coupling of liquid slosh and flexible structure(LF), as well as the parts
that exceed control saturation. The merits of the FDO lie in that estimation error can be arbitrarily small by
adjusting the designed parameters and the prior information is not required. Then, based on the estimation of
FDO, a finite-time TSMC is designed, which has more satisfactory control performance, such as chattering
reduction and fast convergence speed. The stability of the closed-loop system is proved strictly by Lyapunov
theory. Finally, numerical simulations are presented to demonstrate the effectiveness of the proposed method.
INDEX TERMS Spacecraft, flexible vibration, liquid fuel slosh, control saturation, fuzzy disturbance
observer, terminal sliding mode control.
I. INTRODUCTION
Over the past few decades, the space technology develop-
ment is recognized as an important part for national security,
earth observation, planetary exploration and so on. Since
it has been rapidly developed for the control strategy of
rigid spacecraft, many methods have been introduced to
enhance the stability of the control effect, such as sliding
mode control(SMC) [1]–[4], adaptive control [5], [6], robust
control [7]–[9]. However, The missions of the spacecraft are
becoming increasingly complex, flexible appendages [10]
and liquid storage chamber [11] are widely adopted on the
spacecraft, which not only change the control mode but
also make the control more difficult. For example, there
are some features of highly flexibility and low damping for
flexible appendages such as solar panels and long anten-
nas [12]. Hence, the control performance and stability of
LFS-FS would be deteriorated due to the coupling between
the rigid body and the flexible appendages [13]–[15]. In addi-
tion, because the space task becomes more complex, longer
The associate editor coordinating the review of this manuscript and
approving it for publication was Jianyong Yao .
residence time and much more liquid fuel are needed for the
spacecraft [16], [17]. However, due to the sloshing of the
liquid fuel, the maneuverability of spacecraft is inevitably
influenced, which causes the bad control performance and
even the failure task [18], [19]. Therefore, the attitude stabil-
ity control for LFS-FS with model uncertainties and external
disturbance have been one of the important research topics.
What’s more, as is known to all, the control torque pro-
duced by the actuator is usually not infinite in practical
applications, which will no doubt cause the control sat-
uration problems and lead substantial degradation to the
system [20]–[24]. In [25], American researcher RD Robinett
discussed a generalized feedback control law design method
under control saturation constraints as early as 1997. Then
in [26], [27], the Boskovic in Yale taked the control saturation
problem into account in the design of the control law for
spacecraft and designed a control algorithm based on vari-
able structure. In [28], Hu et al. investigated a finite-time
controller under input saturation based on a second-order
disturbance observer and the adaptive control.
Although Boskovicv et al. [26], [27], Hu et al. [21],
Xiao et al. [22], Hu et al. [28], Xiao et al. [29], Hu et al. [30],
149810 This work is licensed under a Creative Commons Attribution 4.0 License. For more information, see http://creativecommons.org/licenses/by/4.0/ VOLUME 7, 2019
L. Dou et al.: FDO-Based SMC for LFS-FS Under Control Saturation
Zou et al. [31], Ruiter [32], and Zhu et al. [33], Xia et al. [34],
Lu and Xia [35] have developed some controllers to effec-
tively deal with the constrained actuator output, hardly any
study discussed the control algorithm for LFS-FS under con-
trol saturation constrain and thus cannot meet the demands
of aerospace task for the future. For the above reasons, this
paper will propose a control method for the stability control
problem of LFS-FS with actuator saturation.
The disturbance observer, is widely used for nonlinear
control problems with external disturbances and uncertain-
ties. Reference [36], [37] proposed extended state observer-
based adaptive and robust control methods, which estimated
the unmeasurable system states and the additive distur-
bances effectively. A novel adaptive sliding mode disturbance
observer is proposed in [38], which could achieve the pre-
cise and fast trajectory tracking control for space manipula-
tor after capturing an uncertain space target. And the FDO
has universal approximation capability for the system with
unknown disturbances and uncertainties [39]. Therefore, it is
widely employed to counteract the system disturbances and
improve controller robustness. A novel control method for
the flight simulator was proposed in [40], in which a FDO
was designed to compensate the external disturbances. The
unknown uncertainty and disturbance of nonlinear system
were estimated by the FDO adopted in [41], [42], and the
simulations are carried out to verify the effectiveness and the
applicability of the FDO for the nonlinear control system with
uncertainty.
Furthermore, in nonlinear control designs, SMC is a well-
known and powerful control scheme that has been widely
used. In [43]–[45] the SMC was adopted to design an effec-
tive and stable controller for nonlinear systems. The hierar-
chical SMC was employed to design the controller for the
space vehicle in [46]. In [6], a second-order SMC method
for spacecraft was presented to guarantee control error can
converge to zero in the finite time. SMC has some advan-
tages such that it is not sensitive to parameters change. And,
the TSMC adds terminal items based on SMC, which makes
the control system have faster convergence speed and ensures
that the system converges in a finite time.
In this work, the integration of FDO and TSMC is proposed
to solve the attitude stability problem for the LFS-FS under
control saturation, uncertainties and disturbances, which can
ensure the control system of the LFS-FS is stable. The main
contributions are summarized as follows:
1) From the theoretic aspect, an FDO-based TSMC strat-
egy is proposed in this paper. The FDO, designed by
the fuzzy logic system (FLS), can approximate the uncer-
tainties and various types of disturbances, which provides
freedom from derivative of disturbance bound assumption.
What’s more, the FDO-based TSMC strategy is proposed
to guarantee the asymptotically stability, and reject the
lumped uncertainty with lower chattering and higher accu-
racy, which has been rigorously proved by the Lyapunov
theory.
FIGURE 1. The diagram of LFS-FS.
2) From the engineering application aspect, the integrated
design of FDO and TSMC can provide the fast and high pre-
cision attitude stabilization control of the complex spacecraft,
which is coupled with liquid slosh and flexible vibration,
even in the presence of external disturbances and control
saturation. What’s more, the simulations and comparison
analysis are presented to verify the effectiveness and better
performance of the proposed method, which can reduce the
chattering and accelerate the convergence speed effectively.
The outline of this paper is as follows. At first, the math-
ematical model of LFS-FS under control input constraint is
introduced in Section II. Then, FLS is introduced and the
designed FDO is adopted to estimate the lumped uncertainty
in Section III. In Section IV, the TSMC based on the esti-
mations of FDO is proposed and the Lyapunov function is
adopted to prove that the controller system is stable. Simula-
tion results and analysis are shown in Section V. The paper
ends with the conclusions in Section VI.
II. PROBLEM FORMULATION
A. MATHEMATICAL SYSTEM MODEL
The diagram of LFS-FS is shown in Figure 1. The dynamic
equations can be described by [16], [47]
˙qv=1
2(q0I3+q×
v)ω
˙q0= − 1
2qT
vω
(1)
(¨χ+C1˙χ+K1χ+N˙ω=0
M0¨η+C2˙η+K2η+M˙ω=0(2)
J˙ω+NT¨χ+MT¨η= −ω×(Jω+NT˙χ+MT˙η)+u+d
(3)
where the unit quaternion q=[q0qT
v]T=[q0q1q2q3]T,
q0is the scalar part of qand qvis the vector part, and satisfies
qT
vqv+q2
0=1; ω∈R3is the angular velocity of the body
fixed frame with respect to the inertia reference frame, χand
ηare the modal coordinate vectors of flexible structure and
liquid slosh respectively; u,drepresent the control torque
and external disturbance, respectively; The inertia matrix J
is described as J=J0+1J, where the nominal part and
the uncertain part are represented by J0and 1J;Ki,Ci(i=
1,2) denote the stiffness and damping matrices, respectively;
M,N∈R4×3are the coupling matrixs of the liquid slosh and
flexible structure between rigid dynamics, respectively. And
the expressions of Mand the quality matrix M0of liquid slosh
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L. Dou et al.: FDO-Based SMC for LFS-FS Under Control Saturation
are given as follows.
M=
0m1b10
−m1b10 0
0m2b20
−m2b20 0
M0=
m10 0 0
0m10 0
0 0 m20
0 0 0 m2
(4)
where mi,bi(i=1,2) represent the quality of the sloshing
liquid mass and the distance between each sloshing mass
and the center mass of the total spacecraft, respectively. The
q×
v∈R3×3is a cross product matrix,which can be defined as
follows.
q×
v=
0−q3q2
q30−q1
−q2q10
(5)
B. CONTROL-ORIENTED MODE L
Consider the situation of actuator input saturation,
the dynamic model (3) is expressed as
J˙ω+NT¨χ+MT¨η= −ω×(Jω+NT˙χ+MT˙η)
+sat(u)+d(6)
where sat(·) represents the nonlinear saturation characteristic
of the actuators, defined as
sat(ui)=
umax,ui>umax
ui,umin ≤ui≤umax
umin,ui<umin
(7)
where umax and umin are the upper bound and lower bound of
actuator torque respectively.
Design a virtual variable 1u=(1u1, 1u2, 1u3)∈R3,
in which 1ui,i=1,2,3 is given as
1ui=
umax −ui,ui>umax
0,umin ≤ui≤umax
umin −ui,ui<umin
(8)
Thus (6) is rewritten as
J˙ω+NT¨χ+MT¨η= −ω×(Jω+NT˙χ+MT˙η)
+u+1u+d(9)
Define C=ω×J0,D=I3and the lumped uncertainty
4= −ω×1Jω−1J˙ω−ω×NT˙χ−ω×MT˙η
−NT¨χ−MT¨η+d+1u
thus we have
J0˙ω= −Cω+Du +4(10)
In this work, our goal is to achieve the attitude stabilization
in the presence of inertia uncertainty, external disturbances
and LF with bounded control input. The main objective is to
design a FDO-based TSMC scheme to guarantee the attitude
stabilization of complex spacecraft system (1) (2) and (9),
which means the qand ωwill converge to an arbitrary small
set containing the origin.
III. FUZZY DISTURBANCE OBSERVER
To give a better understanding of FDO, the FLS is briefly
introduced firstly in this section. After that, we designed a
new fuzzy disturbance observer to estimate the whole uncer-
tainties 4, which could realize an arbitrarily small estimation
error by adjusting the designed parameters.
A. DESCRIPTION OF THE FUZZY LOGIC SYSTEM
The FLS contains five parts: fuzzifier, rule bases, inference
engine, type-reducer, and defuzzifier. The fuzzy inference
engine uses IF-THEN rules to map from an input compact set
X=[x1,x2,· · · ,xn]∈Rnto an output compact set Y∈R.
The fuzzy rules can be described as
Ri:IF x1is Ai
1,and . . . and,xnis Ai
n,then y is Y (11)
where Ai
1,Ai
2,...,Ai
nare fuzzy variables and Yis a fuzzy
singleton.
Using center-average defuzzifier and singleton fuzzifier,
the output of FLS is represented as
y(x)=Pp
j=1hj[Qn
i=1µAj
i
(xi)]
Pp
j=1[Qn
i=1µAj
i
(xi)]
=
p
X
j=1
θjξj(x)=θTξ(x) (12)
where prepresents the number of fuzzy logic rules, θT=
(h1,h2,···,hp)Tis the vector of adjustable parameters,
ξj(x)=Qn
i=1µAj
i
(xi)
Pp
j=1(Qn
i=1µAj
i
(xi)) denotes the vector of fuzzy basis
function, and the µAj
i
(xi) denotes the membership function.
Assumption 1 [48], [49]: Let f (x)be a continuous func-
tion defined on a compact set Mx. Then, for any constant
ε > 0, there exists a fuzzy logic system (12) such that
sup
x∈Mxf(x)−θTξ< ε .
B. FDO DESIGN FOR SPACECRAF T ATTITUDE
CONTROL SYSTEM
In this work, we design the fuzzy disturbance observer for
spacecraft attitude control system. Based on the (10) and (12),
the FDO is constructed as
˙z= −σz+σJ0ω−Cω+Du +ˆ
4(13)
The estimation of the lumped uncertainty 4is given as
ˆ
4=ˆ
θTξ(14)
thus the FDO can be written as:
˙z= −σz+σJ0ω−Cω+Du +ˆ
θTξ(15)
149812 VOLUME 7, 2019
L. Dou et al.: FDO-Based SMC for LFS-FS Under Control Saturation
where zdenotes the observer state, the designed constant
σ > 0, ˆ
4is the estimate value of the lumped uncertainty 4,
ˆ
θis achieved by an adaptation law which is expressed as
˙
ˆ
θ=κ0ξ(e+γ0ε∗) (16)
where the constants κ0>0,γ0>0 are the designed values.
Define the estimated error e=J0ω−z, the dynamics of
the observation estimation errors is described as
˙e=J0˙ω− ˙z= −σe+4−ˆ
4(17)
According to the Assumption 1, 4satisfies the following
form
4=θ∗Tξ+ε(18)
where θ∗is the optimal parameter vector. The εdenotes
approximation error of FLS, which is bounded by ¯ε.
Substituting the (14) and (18) into (17), ˙ecan be rewrit-
ten as
˙e= −σe−˜
θTξ+ε(19)
where ˜
θ=ˆ
θ−θ∗denotes the parameter estimation error.
The eis exponentially convergent, when ˜
θapproaches 0.
Define the disturbance reconstruction error as ε∗=
−˜
θTξ+ε, let m= − ˜
θTξ, then the (19) is transformed into
˙e= −σe+ε∗, and thus we have the reconstructed error
formulated as:
ε∗=σe+ ˙e(20)
Theorem 1: Consider the (9) and the FDO (13). If the
adaptive law of the parameter vector ˆ
θis chosen as (16), and
the parameters κ0>0,γ0>0, then e is uniformly stable
converging to a small region.
Proof: The following Lyapunov function candidate is
chosen:
VF=1
2e2+1
2κ0
˜
θT˜
θ(21)
Taking the time derivative of (21), and combining (19) and
(16), ˙
VFsatisfies:
˙
VF=e˙e+1
κ0
˜
θT˙
ˆ
θ
=e(−σe−˜
θTξ+ε)+˜
θTξ(e+γ0ε∗)
= −σe2+eε+γ0˜
θTξε∗
= −σe2+eε−mγ0(m+ε)
= −σe2+eε−γ0m2−γ0mε(22)
It is noted that the following inequalities satisfy
eε≤1
2σe2+1
2σ¯ε2
mε≤1
2m2+1
2¯ε2(23)
Then, ˙
VFin (22) can be transformed into following form
˙
VF≤ −σe2+1
2σe2+1
2σ¯ε2−γ0m2+γ0
2m2+γ0
2¯ε2
= − 1
2σe2+1
2σ¯ε2−γ0
2m2+γ0
2¯ε2(24)
Integrating both sides of (24) from 0 to Tyields
1
2ZT
0
σe2dt +1
2ZT
0
γ0m2dt
≤V(0) −V(T)t+ZT
0
1
2σ¯ε2d+ZT
0
γ0
2¯ε2dt
≤V(0) +ZT
0
(1
2σ+γ0
2)¯ε2dt (25)
which is equivalent to
ZT
0
σe2dt +ZT
0
γ0m2dt
≤e2(0) +1
κ0
˜
θ(0)T˜
θ(0) +ZT
0
(γ0+1
σ)¯ε2dt (26)
The robust performance of (26) can be explained by
Barbalat’s lemma [50]. If ε∈L2, i.e. R∞
0ε2dt <∞, then
e∈L2and m∈L2. This means limt→∞ kek=0 and
limt→∞ kmk=0. Even though ε /∈L2,e2is bounded by ε2.
Therefore, the observation error can be reduced to any small
value with adjusting the σand γ0. Thus, ˆ
4can estimate 4
with arbitrarily small error.
IV. COTROLLER DESIGN FOR SPACECRAFT SYSTEM
A. FDO-BASED TSMC DESIGN
In the subsection, a FDO-based terminal sliding mode con-
troller is designed for the LFS-FS with inertia uncertainty and
external disturbance under actuator input saturation, which
can guarantee the stabilization of the spacecraft system with
high speed and precision.
The following sliding mode surface is designed:
s=ω+k·β(qv) (27)
where s=[s1s2s3]T∈R3,k>0 is user-designed
constant, and β(qv)=[β(q1)β(q2)β(q3)]Tsatisfies the
following form:
β(qi)=(sigr(qi)if ¯si=0or ¯si6= 0,|qi|> ν
a11qi+a12 sig2(qi)if ¯si6= 0,|qi|≤ν(28)
where ¯si=ωi+k·sigr(qi), sigr(qi)=|qi|rsign(qi), i=
1,2,3, 0 <r<1, ν > 0 is a small constant, and a11 and a22
satisfy a11 =(2 −r)νr−1,a22 =(r−1)νr−2.
Then, base on the (1) and (3),the time derivative of (27) is
obtained as
J0˙s= −ω×(J0ω+NT˙χ+MT˙η)+k·J0˙
β(qv)
−NT¨χ−MT¨η+u+1u+d−1J˙ω−ω×1Jω
=g(t)+u+4(29)
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L. Dou et al.: FDO-Based SMC for LFS-FS Under Control Saturation
where 4denotes the lumped uncertainty of the system, g(t)
denotes the normal part of the system, which satisfies the
following form
g(t)= −ω×J0ω+k·J0˙
β(qv) (30)
˙
β(qi)=(r|qi|r−1˙qi,if ¯si=0or ¯si6= 0,|qi|> ν
a11 ˙qi+2a12 |qi|˙qi,if ¯si6= 0,|qi|≤ν
(31)
Thus, the uis designed as:
u= −g(t)−δ1s−δ2sigr(s)−ˆ
θTξ(32)
where δ1,δ2are positive designed constants.
Lemma 1 [51]:Consider nonlinear system ˙x=f(x,u),
where x is the state vector and u is the control vector. Assum-
ing that there are continuous differentiable positive definite
functions V (x), scalar λ > 0,0<a<1,0< ϑ < ∞,
which makes inequality ˙
V(x)≤ −λVα(x)+ϑvalid, then the
system ˙x=f(x,u)is actually stable in finite time, and the
convergence time is
Treach ≤V1−α(x0)
λφ(1 −α),0< φ < 1 (33)
where V(x0) is the initial value of V(x).
B. STABILITY ANALYSIS
Theorem 2:Consider the nonlinear sapcecraft system (1)-(2)
with control saturation (6). If designing the terminal sliding
mode controller (32)augmented by the fuzzy disturbance
observer (13) and adaptive law (18)under κ0, γ0>0,
0<r<1, and δ1, δ2>0, then the attitude quaternion
qvand the attitude angular veloticy ωwill converge to small
neighborhoods of origion by selecting suitable controller
parameters.
Proof: Define Lyapunov function candidate as
V=V1+V2+V3(34)
where
V1=1
2sTJ0s(35)
V2=1
2q2
v(36)
V3is (21) which verifies the stability of the FDO. The math-
ematical proof has been shown in section III-B.
Taking the time derivative of (35), it follows that
˙
V1=sTJ0˙s
≤sT[g(t)+u+]
=sT[−δ1s−δ2sigr(s)−ˆ
θTξ+θ∗Tξ+ε]
= −δ1s2−δ2
3
X
i=1
|si|r+1−sT˜
θTξ+sTε
= −δ1s2−δ2
3
X
i=1
|si|r+1+sTm+sTε(37)
By applying the following inequalities
sTm≤1
2σs2+1
2σm2
sTε≤1
2σs2+1
2σ¯ε2(38)
Thus, the derivative ˙
V1can be derived as
˙
V1≤ −δ1s2−δ2
3
X
i=1
|si|r+1+1
2σs2+1
2σm2+1
2σs2
+1
2σ¯ε2
≤ −(δ1−σ)s2−δ2
3
X
i=1
|si|r+1+1
2σm2+1
2σ¯ε2
≤ −(δ1−σ)s2−2(r+1)/2δ2
λ(r+1)/2
max
V(r+1)/2
1+1
2σm2+1
2σ¯ε2
(39)
When the parameters are chosen as δ1> σ > 0,
the inequality (39) becomes
˙
V1≤ − 2(r+1)/2δ2
λ(r+1)/2
max
V(r+1)/2
1+1
2σm2+1
2σ¯ε2(40)
According to the Lemma1, it means that the system is
actually stable in finite time and the system states will reach
the sliding surface s=0 (27) in finite time, which implies
that
2(q0I3+q×
v)−1˙qv= −kβ(qv) (41)
thus we can draw the conclusion that ˙
V2=qv˙qv<0, the qv
is exponentially stable. This means that the system states will
reach to the desired equilibrium point asymptotically under
the designed control law. Therefore, the proof is completed.
V. SIMULATION RESULTS AND ANALYSIS
In this section, the effectiveness and performance of the FDO
(31) and TSMC in (32) for LFS-FS (1)(2)(6) will be verified
by simulation.
A. PARAMETER SETTING
It is assumed that the inertia matrix and nominal inertia matrix
for the coupled spacecraft are selected as
J=[360 3 4;3 279 10;4 10 198] (42)
J0=[350 0 0;0 270 0;0 0 190] (43)
The physical parameters are chosen as [47], [48],
[52]:
N=
6.45637 1.27814 2.15629
−1.25819 0.91756 −1.67264
1.11687 2.48901 −0.83674
1.23637 −2.6581 −1.12503
C1=diag{0.0086,0.0190,0.0487,0.1275 }
K1=diag{0.5900,1.2184,3.5093,6.5005 }
C2=diag{3.334,3.334,0.237,0.237 }
K2=diag{55.21,55.21,7.27,7.27 }
149814 VOLUME 7, 2019
L. Dou et al.: FDO-Based SMC for LFS-FS Under Control Saturation
FIGURE 2. Time responses of quaternions in case 1. (a) Without control
saturation.(b) With control saturation.
m1=20kg,m2=0.8kg,b1=1.127m,b2=0.994m.
The controller parameters of controller design are set to be:
δ1=100, δ2=2, k=0.2, r=0.8, σ=1.2,
κ0=γ0=1. The initial values of angular velocity and
attitude for coupled spacecraft are ω=−0.01 0.02 0.03
and q=[0.8832 0.3−0.2−0.3]. The external dis-
turbance added to the system are designed as d=(kωk2+
0.05)[ sin 0.8tcos 0.5tcos 0.3t]T.
B. SIMULATION ANALYSIS
To further illustrate the effectiveness of the proposed method,
simulation of two cases are made for the comparison. In the
first case, the robustness of the proposed control scheme is
verified by applying it to the conditions with input saturation
and without input saturation respectively. In the second case,
to prove the necessity of studying LFS-FS under control
saturation, the adaptive variable structure control presented
in [21] is applied to the control input saturation problem
considered above, the simulation results show the advantages
of the controller in this paper.
Case 1: In this case, we verify that the proposed algorithm
can effectively overcome the drawback caused by the control
saturation. Firstly, the angular velocity and attitude quater-
nion trajectories are shown in (a) of Figure 2 and Figure 3.
FIGURE 3. Time responses of angular velocity in case 1. (a) Without
control saturation. (b) With control saturation.
As we can see, the coupled spacecraft system achieves sta-
bilization smoothly in less than 20 s with a high accuracy of
10−3in static error. The trajectories of the control input is
shown in Figure 4 (a), which demonstrates the upper bounder
of required torque is aboud 10Nm. Besides, Figure 5 (a) shows
the effectiveness of the FDO. The reconstructed lumped
uncertainty is caught almost immediately.
Then, to testify the robustness of the proposed controller
when faced with input constraint conditions. The restrictions
on the actuator output torque are set as umax =2Nm and
umin = −2Nm, which are much smaller than the required
ones. Comparing the (a) and (b) of Figure 2 and Figure 3,
it can be seen that the state trajectory and the convergence
time for and are substantially unchanged, and the static error
and the vibration are kept within a small scale, indicating
that the controller is robust to the input limitation and can
effectively overcome the lag of control effect caused by it.
Time responses of the demand control torques are depicted
in Figure 4 (b). Note that the control torques are strictly
limited to [−2,2]Nm.Figure 5 shows the result of FDO which
reflects that with the consideration of the constrained part
of the control input, the reconstructed lumped uncertainties
became larger in the initial stage, but when the system is
gradually stable under the control, the observer’s value would
be restored to a small one.
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FIGURE 4. Time responses of control torques in case 1. (a) Without
control saturation. (b) With control saturation.
Case 2: In this case, the fuzzy disturbance observer
based terminal sliding mode controller (FDOTSC) proposed
in (32) is compared to the adaptive variable structure con-
troller(AVSC) [21]. The red solid line and blue dash line
in Figure 6-Figure 10 denote the control performance by
(32) and AVSC in [21] respectively. For the AVSC in [21],
the control parameters are chosen as δ=0.02, r =0.05. The
parameters in (32) are consistent with case 1. Figure 6 and
Figure 7 show time responses of LFS-FS attitude quaternion
and the angular velocity. As we can see, with the controller in
(32), the system achieves stabilization smoothly with the time
less than 25 s and with a high accuracy of 10−3in static error.
Although with the AVSC more time is required to achieve
the attitude stabilization and the static error exists even after
100 s. Accordingly, the case of FDOTSC in (32) achieves
the better performance than the AVSC in [21], which further
illustrates the advantages and feasibility of the FDOTSC
method.
Time responses of control torques is pictured in Figure 8.
The demand control is strictly limited in [-2Nm,2Nm] for
both of the controller. Furthermore, note that when the space-
craft system stabilize, the control torque in FDOTC is equal to
the lumped uncertainties observed by the FDO, which com-
pensates for the system and makes the performance better.
FIGURE 5. Time responses of FDO in case 1. (a) Without control
saturation. (b) With control saturation.
FIGURE 6. Comparison of angular velocity.
Because the first two orders of vibration and sloshing
modes have the greatest impact on system stability, we just
list the time responses of the sloshing liquid η1,η2and flexi-
ble structure χ1,χ2, which is shown in Figure 9 and Figure 10.
Although, sloshing liquid and flexible structure can cause
tracking error and instability during spacecraft maneuvering
149816 VOLUME 7, 2019
L. Dou et al.: FDO-Based SMC for LFS-FS Under Control Saturation
FIGURE 7. Comparison of quaternion.
FIGURE 8. Comparison of control torque.
process, the FDOTSC compared to AVSC in [21] can effec-
tively track and observe the effects of flexible vibration and
liquid sloshing, and achieve fast and stable attitude tracking.
From the Figure 9 and Figure 10, as we can see, compared
with [21], the FDOTSC proposed by this paper brings less
liquid sloshing and flexible vibration. One reason is because
the FDO with universal approximation can observe the over-
all disturbance including the inertia uncertainty, external dis-
turbance and the coupling of LF, another reason is that the
FDO-based TSMC controller designed in this work improves
system convergence speed and control performance. And the
figures also verifies the effectiveness of the proposed method
in this work for the problem of liquid slosh and flexible
vibration.
FIGURE 9. Comparison of sloshing liquid η.
FIGURE 10. Comparison of flexible structure χ.
Summarizing all the cases above, the controller (FDOTSC)
proposed is robust to the control saturation, besides, the flex-
ibility in the selection of control parameters can be used to
achieve better performance while satisfying the constraints of
control torque and uncertainty. In addition,the control method
AVSC designed for spacecraft with small rigid body cannot
control the spacecraft with as well. Thus, the studying of
control method for FLS under control saturation is of strong
utility value. This control method provides a theoretical basis
for the practical application of advanced control theory in the
design of spacecraft control system.
VI. CONCLUSION
In this paper, the integration of TSMC and FDO for the
LFS-FS under control saturation, uncertainties and distur-
bances is proposed. The mathematical expressions for the
LFS-FS under control saturation constrain is firstly intro-
duced and the FDO is designed to estimate the lumped uncer-
tainties caused not only by the external disturbance but also
by the fuel slash, flexible structure, inertia uncertainties and
control saturation. With the estimated value by the FDO,
the TSMC is presented, which makes up for the shortcomings
of traditional sliding mode control and ensures that the system
converges in a finite time. The stability of the closed-loop
system is proved by Lyapunov theory. Simulations of the
two cases are carried out to testify the performance of the
proposed control scheme. However, the directly inhibition of
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L. Dou et al.: FDO-Based SMC for LFS-FS Under Control Saturation
the flexible vibration and the fluid shaking is not considered
in this paper, which is one of subjects for the improvement of
control methods for LFS-FS.
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LIQIAN DOU received the B.S., M.S., and Ph.D.
degrees in automatic control from Tianjin Uni-
versity, Tianjin, China, in 1999, 2005, and 2008,
respectively. He was an Academic Visitor with the
School of Electrical and Electronic Engineering,
University of Manchester, Manchester, U.K., from
June 2015 to June 2016. He is currently an Asso-
ciate Professor with the School of Electrical and
Information Engineering, Tianjin University. His
main research interest includes coordinate control
of multi-UAVs.
MIAOMIAO DU received the B.S. degree in
Electrical and Information Engineering from
Zhengzhou University, in 2017. She is currently
pursuing the master’s degree with the School
of Electrical and Electronic Engineering, Tianjin
University. Her main research interests include
nonlinear control and flight control.
XIUYUN ZHANG received the B.S. degree in elec-
trical engineering and automation from Qingdao
University, in 2014. She is currently pursuing the
Ph.D. degree with the School of Electrical and
Information Engineering, Tianjin University. Her
main research interests include fault diagnosis and
fault tolerant control for spacecraft systems.
HUI DU received the M.S. degree from the School
of Electrical and Information Engineering, Tianjin
University, in 2019. Her main research interests
include fault tolerant control and flight control.
WENJING LIU received the Ph.D. degree in auto-
matic control from Tianjin University, Tianjin,
China, in 2009. She is currently a Senior Engineer
with the Beijing Institute of Control Engineering.
Her main research interests include fault diagno-
sis and fault-tolerant control for satellite control
systems.
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