![](https://csdnimg.cn/release/download_crawler_static/18883179/bg5.jpg)
Therefore, the preceding fault types can be modeled in the following form
u
F
i
= b
i
u
i
+ α
i
(4)
where u is the desired torque signal of the ith reaction wheel with i = 1, 2, . . . , n, 0 ≤ b
i
≤ 1 represents
the effectiveness factor of the ith reaction wheel, α
i
is the uncertain faulty input which entering the
actuators in an additive form and τ
i
is the actual torque exerted on the satellite system. In addition,
the relationship between actuator fault types and the model parameters are demonstrated in Table 1.
Fault Type b
i
α
i
F1 Loss of partial effectiveness 0 < b
i
< 1 0
F2 Bias fault 1 α
i
= 0
F3 Outage fault 0 0
Table 1: Types of actuator faults
In this paper, the result mainly discusses loss of partial effectiveness fault(F1), bias fault(F2) and
outage fault(F3). For the simplicity of this study , it is assumed that every actuator only experiences
one kind of fault.
Substracting (4) into (3), the faulty dynamic equation can b e described by,
˙ω = −J
−1
ω
×
Jω + J
−1
Dbu + J
−1
Dα + J
−1
d (5)
where b = diag{b
1
, . . . , b
n
} is regarded as the actuator effectiveness matrix and α = [α
1
, . . . , α
n
]
T
is
the additive actuator fault vector.
It is well known that the bias fault, loss of effectiveness fault and outage fault of reaction flywheel often
happen in the attitude systems of satellite, the severe working environment of satellite demands that
the control scheme could handle with various fault timely. The time and magnitude of fault occurred
in attitude systems are unknown, thus increase the difficulty of FTC design approach. In generally,
an active FTC scheme includes two parts: fault diagnosis module and fault tolerance control design
module. In order to design an active FTC approach, the main objectives of this paper are described
by the following
• Design a nonlinear sliding mode fault detection observer for the satellite dynamic equation to
monitor the occurrence of unknown actuator faults.
• Design an adaptive sliding mode fault estimation observer for the satellite dynamic equation to
obtain the accurate estimation value of actuator fault occurred in the attitude systems.
• Design an active FTC scheme for the attitude systems of rigid satellite in actuator saturation
and multiple faults case to ensure that all the signals of the closed-loop attitude systems are
asymptotically stable in a finite time.
The whole active FTC scheme developed in this paper can be shown in Figure 1. To achieve the
objectives described above, we recall the following Assumptions and Lemmas, which will be needed in
the proof of the main results.
4