近空间高超声速飞行器故障容错控制的T-S模糊系统
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"这篇研究论文探讨了Takagi-Sugeno (T-S)模糊系统的容错控制问题，特别是在近空间高超音速飞行器存在执行器故障的背景下。作者提出了一个综合时间变化偏置故障和时间变化增益故障的通用执行器故障模型。接着，设计了滑模观测器（SMOs）来生成一系列表征故障检测和隔离的残差。基于Lyapunov稳定性理论，提出了一种新颖的故障诊断算法，该算法能估计执行器故障，而无需像某些现有工作那样假设输出误差的时间导数已知。此外，还设计了一个故障估计观察器。利用估计出的执行器故障，提出了一种补偿故障影响的适应方案。论文进一步给出了根据Lyapunov稳定性理论推导的SMO存在的充分条件。"
在T-S模糊系统中，容错控制是确保系统即使在部件出现故障时仍能保持稳定运行的关键技术。T-S模糊系统是一种广泛应用的非线性系统建模方法，通过一组线性子系统和模糊规则来描述复杂的非线性动态行为。论文首先建立了一个全面的执行器故障模型，考虑了故障可能随着时间动态变化的情况，这使得模型更具普适性。
滑模观测器（SMOs）是一种实时故障检测工具，通过对系统状态的估计来监测系统的异常行为。在这里，设计的SMOs能够生成用于故障检测和隔离的残差信号。通过分析这些残差，可以判断系统是否存在故障以及故障的位置。
故障诊断算法是基于Lyapunov稳定性理论提出的，该理论是控制理论中的基础工具，用于证明系统的稳定性。新算法克服了传统方法对输出误差时间导数的先验知识依赖，提高了故障诊断的灵活性和准确性。
然后，论文开发了一个故障估计观察器，其目的是更准确地估计执行器的故障状态，以便更有效地补偿故障影响。通过这个适应性方案，系统可以调整自身以抵消故障导致的性能下降。
最后，论文提供了SMO存在的一个充分条件，这是设计SMO的重要步骤，确保了在实际应用中观测器的可行性。这一条件基于Lyapunov稳定性理论，保证了系统在故障条件下仍能保持稳定。
这篇论文为T-S模糊系统的容错控制提供了一个全面且创新的框架，特别适用于近空间高超音速飞行器等复杂系统的故障管理，对实际工程应用具有重要的理论指导价值。
654 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 20, NO. 4, AUGUST 2012
The control objective under normal conditions is to design a
proper state feedback controller u(t) such that the system (2) is
stable.
However, in practical application, actuators may become
faulty. Bias faults and gain faults are two kinds of actuator
faults that are commonly occurring in practice. An actuator bias
fault can be described as
u
f
i
(t)=u
i
(t)+f
i
(t),i=1,...,m, (6)
where f
i
(t) denotes a bounded signal, and an actuator gain fault
can be described as
u
f
i
(t)=(1−ρ
i
(t))u
i
(t),i=1,...,m, (7)
where 0 ≤ ρ
i
(t) ≤ 1, which is supposed to be unknown, de-
notes the remaining control rate. Therefore, the two previously
mentioned kinds of actuator faults can be uniformly described
as
u
f
i
(t)=(1−ρ
i
(t))u
i
(t)+f
i
(t). (8)
Furthermore, a more general fault model can be given as
u
f
i
(t)=(1−ρ
i
(t))u
i
(t)+
p
i
j=1
g
i.j
f
i,j
(t) (9)
where f
i,j
(t),i=1,...,m, j =1,...,p
i
, denotes a
bounded signal, p
i
is a known positive constant, and g
i,j
denotes an unknown constant. With no restriction, let us sup-
pose p
1
= p
2
= ···= p
m
= p, with p being a known positive
constant. Consider the following notation: a
i,j
(t)=g
i.j
f
i,j
(t).
Then, (9) can be rewritten as follows:
u
f
i
(t)=(1−ρ
i
(t))u
i
(t)+
p
j=1
a
i,j
(t). (10)
Denote
Γ(t) = diag(ρ
1
(t),...,ρ
m
(t)) (11)
F (t)=[f
1
,f
2
,...,f
m
]
T
,f
i
=
p
j=1
a
i,j
(t). (12)
Then, we have
u
f
(t)=(I − Γ(t))(u(t)+F (t)),t≥ t
j
(13)
where the failure time instant t
j
is unknown, and I denotes the
identity matrix with appropriate dimensions. In this paper, both
bias and gain faults are handled by considering the general fault
model (13).
Notice that, in the following, just for the sake of notational
simplicity, we will use h
i
,ρ
i
, and a
i,j
to denote h
i
(z(t)),ρ
i
(t),
and a
i,j
(t), respectively.
Now, the control objective is redefined as follows. An active
FTC approach is proposed to make the system (2) stable under
normal and faulty conditions. Under normal condition (no fault),
a state feedback control input u(t) is designed such that the
system (2) is stable. Meanwhile, the FDI algorithm is working.
As soon as an actuator fault is detected and isolated, the fault-
estimation algorithm is activated. The obtained fault estimation
is used to design a proper control input u(t) such that the system
(2) is still maintained stable under faulty case.
Remark 1: In the literature, many papers consider actuator
faults. However, most of them only considered bias faults. Gain
faults have not attracted enough attention. In [11], a class of bias
fault was studied, where the fault was assumed to be an unknown
constant. However, in practicalapplication, thefault may betime
varying. Equation (10) is a deterministic but uncertain actuator
model which represents a class of practical actuator faults such
as actuator gain variations and measurement errors. In fact, the
fault model in [9]–[11] can be described by (10). If ρ
i
(t)=0,
model (10) then becomes the bias fault model that is considered
in [9] and [10]. If ρ
i
(t) is an unknown constant and f
i
(t)=0,
then model (10) denotes the constant bias faults model as in [11].
Hence, the proposed actuator fault model (10) is more general
and has wider practical use than the classical ones.
III. F
AULT DIAGNOSIS AND ACCOMMODATION
In this section, the main technical results of this paper are
given. We will first formulate the fault-diagnosis and accommo-
dation problem of the aforementioned T–S fuzzy system. We
will then design a bank of sliding-mode observers (SMOs) to
generate residuals, investigate the FDI algorithm that is based
on the SMOs, and propose an FTC scheme to tolerate the fault
using estimated fault information.
A. Preliminary Description
Consider the T–S fuzzy faulty system that is described in (2).
We assume that only actuator faults occur, and no sensor fault
is involved. For simplicity, we consider the case that only one
single actuator is faulty at one time. The actuator fault-diagnosis
problem is formulated as follows: With the available output y,
we propose an observer-based scheme to identify the faulty
actuator, and then estimate the fault.
To solve the problem, we will design a bank of SMOs with
desired actuator fault-detection and fault-estimation properties.
Thus, the following assumptions are made in this paper.
Assumption 1: Matrix B
i
is of full column rank, and the pair
(A
i
,C
i
) is observable.
Assumption 2: There exist known positive constants ¯ρ
i
,
¯
¯ρ
i
,
¯ρ
1
, ¯ρ
2
such that |ρ
i
(t)|≤¯ρ
i
, |˙ρ
i
(t)|≤
¯
¯ρ
i
, ¯ρ
1
= max{¯ρ
1
,
¯ρ
2
,...,¯ρ
m
}, ¯ρ
2
= max{
¯
¯ρ
1
,
¯
¯ρ
2
,...,
¯
¯ρ
m
}, i =1,...,m.
Assumption 3: There exist known positive constants
¯a
1
, ¯a
2
, ¯a
i,j
,
¯
¯a
i,j
such that |a
i,j
(t)|≤¯a
i,j
, |˙a
i,j
(t)|≤
¯
¯a
i,j
, ¯a
1
=
max{¯a
1,1
,...,¯a
i,p
,...,¯a
m,1
,...,¯a
m,p
}, ¯a
2
=max{
¯
¯a
1,1
,...,
¯
¯a
i,p
,...,
¯
¯a
m,1
,...,
¯
¯a
m,p
}, i =1,...,m, j =1,...,p.
Our actuator fault-diagnosis and accommodation scheme
consists of FDI and FTC. We first design the fault-diagnosis ob-
server utilizing SMOs to detect, isolate, and estimate the fault,
and then propose an FTC method to compensate the fault.
B. Fault Detection
In order to detect the actuator faults, we design a fuzzy state-
space observer for the system (8), which is described as follows.
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