Adaptive fuzzy control of unknown nonlinear systems with actuator failures for robust output tracking

Abstract

In this paper, an adaptive fuzzy approach is proposed to deal with robust output tracking of unknown nonlinear systems with actuator failures. The actuator failures under consideration can be both of lock-in-place and loss of effectiveness. By incorporating fuzzy logic approximation, adaptive algorithm and attenuation technique to our design, a fault tolerant control law is developed to guarantee desired output tracking of the controlled system to the given reference model as well as the closed-loop stability, despite there are unknown actuator failures and large uncertainties in the system. A numerical simulation example illustrates the effectiveness of the proposed control approach.

Full text

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Adaptive Fuzzy Control of Unknown Nonlinear Systems with Actuator
Failures for Robust Output Tracking
Ping Li and Guang-Hong Yang
Abstract—In this paper, an adaptive fuzzy approach is
proposed to deal with robust output tracking of unknown
nonlinear systems with actuator failures. The actuator failures
under consideration can be both of lock-in-place and loss
of effectiveness. By incorporating fuzzy logic approximation,
adaptive algorithm and attenuation technique to our design,
a fault tolerant control law is developed to guarantee desired
output tracking of the controlled system to the given reference
model as well as the closed-loop stability, despite there are
unknown actuator failures and large uncertainties in the system.
A numerical simulation example illustrates the effectiveness of
the proposed control approach.
I. INTRODUCTION
With the development of modern industry, lots of systems
are becoming more and more advanced and complicated.
However, this also makes it more possible that some faults
may occur in these systems. As actuator faults may cause
undesired system behavior and sometimes lead to instability
or even catastrophic accidents, it is very important to develop
fault tolerant control (FTC) approaches that would achieve
control objective in spite of actuator faults. Adaptive control
has been used widely in linear systems to accommodate
actuator faults. For example, lock-in-place (stuck at an un-
known value) failures were compensated based on matching
conditions in [1], loss of effectiveness was studied in [2]-[4]
in the framework of linear matrix inequality and multiple
simultaneous actuator failures were dealt with in [5]-[6]
using Multiple Models, Switching, and Tuning method.
However, most real-life systems are nonlinear in nature.
So, nonlinear adaptive FTC has attracted much attention. Tao
et al also developed adaptive control for nonlinear systems
with actuator failures in [1]. And [7] extended its results to
multiple-input-multiple-output systems based virtual group-
ing. However, only feedback linearizable and parametric-
This work is supported in part by Program for New Century Excellent
Talents in University (NCET-04-0283), the Funds for Creative Research
Groups of China (No. 60521003), Program for Changjiang Scholars and
Innovative Research Team in University (No. IRT0421), the State Key
Program of National Natural Science of China (Grant No. 60534010), the
Funds of National Science of China (Grant No. 60674021) and the Funds
of PhD program of MOE, China (Grant No. 20060145019), the 111 Project
(B08015).
Ping Li is with the College of Information Science and Engineering,
Northeastern University, Shenyang, 110004, P.R. China. She is also with
the Key Laboratory of Integrated Automation of Process Industry, Ministry
of Education, Northeastern University, Shenyang 110004, China. Email:
pingping−1213@126.com
Guang-Hong Yang is with the College of Information Science
andEngineering,NortheasternUniversity,
China. He is also with the Key Laboratory of Integrated Automa-
tion of Process Industry, Ministry of Education, Northeastern Uni-
versity, Shenyang 110004, China. Corresponding author. Email:
yangguanghong@ise.neu.edu.cn
Shenyang, 110004,P.R.
strict-feedback systems were considered. Boskovic consid-
ered multiple simultaneous actuator failures for known non-
linear systems in [8]. [9] proposed fault accommodation
approach for systems with Lipschitz nonlinearities. However,
the related results have not appeared in nonlinear systems
whose nonlinearities are completely unknown.
With the development of artificial intelligence, fuzzy logic
and neural network have been introduced into fault detec-
tion and accommodation. Polycarpou presented a general
framework for constructing automated fault diagnosis and
accommodation architectures using on-line approximators in
[10], he furthered his research in [11] based on a neural
network to fault tolerant control of nonlinear flight control
systems. [12]-[15] provided FTC shemes using fuzzy logic
systems (FLSs) or neural networks (NNS), however these
results were obtained on the basis of fault detection and
diagnosis (FDD) mechanism. Since [16] proved that FLSs
are universal approximators and [17] gave the stable adaptive
fuzzy control design for unknown nonlinear systems, many
researchers studied nonlinear systems using adaptive fuzzy
systems to approximate the unknown functions. Though [18]
presented fault tolerant adaptive fuzzy control for a turbine
engine, there is few results on adaptive fuzzy control of sys-
tems with unknown nonlinearities and external disturbances
to accommodate actuator faults without FDD.
This paper studies fault tolerant tracking control for un-
known nonlinear systems with external disturbances against
actuator faults. The main properties compared with the
existing results are that: first, a novel adaptive fuzzy FTC
scheme without FDD is prosed, so the undesired system
behavior caused by false or omitted alarm of FDD mech-
anism is avoided; second, external disturbance is attenuated
effectively to achieve robustness besides the fault tolerant
ability of the closed-loop system; third, our design can
broaden the tolerable fault set by allowing any combination
of lock-in-place and loss of effectiveness happen only if the
system is still controllable. The difficulty in the design is
that the system functions are not continuous because of the
occurred faults, so piecewise analysis is used.
The rest of this paper is organized as follows. Section
II formulates the problem first. Section III introduces the
proposed adaptive fuzzy fault tolerant control scheme. In
Section IV, a numerical simulation example illustrates the
effectiveness of the control method. Finally, Section V con-
cludes this paper.
2008 American Control Conference
Westin Seattle Hotel, Seattle, Washington, USA
June 11-13, 2008
FrB17.6
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II. PROBLEM FORMULATION:
Consider the following nonlinear plant
˙ xn= f(x)+
m
∑
i=1gi(x)ui+d
y = x1
(1)
where x=(x1, x2, ··· , xn)T=(x1, ˙ x1, ··· , x(n−1)
is the state vector, U is a compact set in Rn, y ∈ R is the
output, and ui∈R is the control input which may fail during
operation, i=1, 2, ··· , m, m≥2 provides some redundancy
of the actuator so that when some of them are failed, the
remaining part can still drive the system stable and achieve
acceptable performance. This is reasonable because for some
practical systems, the control surfaces can be divided into
several individually actuated segments. For example, the
aileron segments of an aircraft provide some redundancy
needed for failure compensation. f(x) ∈ R and gi(x) ∈ R
are unknown nonlinear smooth functions. d ∈ R denotes the
external disturbance which is unknown but bounded. So we
cannot get the model of the controlled plant.
The failure model under consideration for fault tolerant
control of the system (1) is
1
)T∈U ⊆Rn
uF
i= ρiui,
ρi∈ [ρi, ¯ ρi],
i ∈ {1,2,··· ,m}
(2)
and
uF
j= ¯ uj,
j ∈ {1,2,··· ,m}
(3)
In this failure model, (2) describes the fault of loss of
effectiveness, where ρidenotes the percentage of the remain-
ing effective part of the corresponding actuator, 0 < ρi≤ 1,
0< ¯ ρi≤1, are the lower and upper bounds of ρirespectively,
ρi≤ ¯ ρi. If ρi= ¯ ρi= 1, there is no failure occurred, i.e. the
actuator is normal. For (2), ρi=0 which means the complete
loss of effectiveness is not considered since it is included in
(3) for ¯ uj= 0. (3) describes the lock-in-place (stuck at an
unknown value) failure. If (3) has occurred, the actuator must
completely lose the effectiveness, thus the control input has
no impact on the controlled system, but the actuator failure
will bring some disturbances if ¯ uj?= 0. Actuators can be
failed as (2) or (3), also, both (2) and (3) may occur during
operation.
The reference model is:
˙ xm= Amxm+Bmr
ym=Cmxm
(4)
where
Am=



0
0
1
0
0 ··· 0
1 ··· 0
··· ···
0 ··· 0
0
0
001
−kn−kn−1··· −k2−k1
Bm= [ 0 ··· ···0 1]T∈ Rn×1
Cm= [1 0 ··· ···0 0] ∈ R1×n



∈ Rn×n,
(5)
k1, ··· ,kn are constants designed such that sn+k1s(n−1)+
···+k(n−1)s+knis a Hurwitz polynomial.
The objective for actuator fault tolerant control is to use
feedback control design for the plant (1) with the actuator
failures (2) or/and (3), to guarantee that all signals in the
closed-loop system are bounded and robust tracking the
output of the given reference model ym. Here the “robust
tracking” refers to attenuate the fuzzy approximation error
and the external disturbance to a prescribed level η. In order
to accomplish this task, the following basic assumption for
the actuator fault tolerant control problem is needed.
Assumption 1: The plant (1) is so constructed that for any
p actuators fail as (3), 0 ≤ p ≤ m−1, and all the other(s)
may lose effectiveness as (2), the remaining effective part of
the actuators can still achieve the desired control objective.
Remark 1: From Assumption 1 we can see that, if only
failure (2) occurred in the system, it allows no redundancy of
the actuator; but as long as failure (3) may occur there must
be some redundant actuators for fault-tolerant. Here both (2)
and (3) are taken into account, so we make m ≥ 2.
III. ADAPTIVE FUZZY FAULT TOLERANT CONTROL
In this section, the design of the fault tolerant control using
adaptive fuzzy approach for plant (1) with actuator failures is
presented. Inspired by [1], a specific structure is considered
in order to obtain the closed-loop stability and the robust
output tracking when the system model and the actuator
failures are all uncertain, here we respect to the nonlinear
system under control is unknown and with disturbance; the
failure style (which actuator has failed and as which failure
model), the failure value (ρiand ¯ uj), and the failure time (at
which the failure occurred) are all unknown.
First we rewrite system (1) in its equivalent matrix form:
˙ x = Ax+B(f(x)+gT(x)u+d)
y =Cx
(6)
where
(u1, u2, ··· , um)T,
g(x) = (g1(x),
g2(x),··· ,
gm(x))T,
u =
A =



0 1 0 ··· 0 0
0 0 1 ··· 0 0
··· ···
0 0 ··· 0 0 1
0 0 ··· 0 0 0



∈ Rn×n,
B = Bm,
C =Cm
(7)
Consider the actuator failures described as (2) and (3), the
actual control input vector u can be expressed as
u = ρυ(t)+σ(¯ u−ρυ(t))
(8)
where υ(t) = [υ1(t), υ2(t), ··· , υm(t)] is the applied control
to be designed, and
ρ = diag{ρ1, ρ2, ··· , ρm},
σ = diag{σ1, σ2, ··· , σm},
?
0
otherwise
¯ u = [¯ u1, ¯ u2, ··· , ¯ um]T
σj=
1
if the jth actuator fails as (3)
(9)
Note that ρυ(t) includes the case where the actuator has no
failure when the corresponding ρi, i ∈ {1, 2, ··· , m}, equals
to 1.
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With the actual input (8) in the presence of actuator
failures (2) and/or (3), the plant (6) can be rewritten as
˙ x = Ax+B(f(x)+gT(x)σ ¯ u+gT(x)(I−σ)ρυ +d),
y =Cx
(10)
where I = diag{1, ··· , 1}m×m. Here a specific proportional
actuation structure is used,
υi(t) = biυ0(t)
i = 1, ··· , m
(11)
where bi is a nonzero constant, υ0(t) is the control signal
needs to be designed.
With the proportional actuation structure (11), (10) is
equivalent to the following form
˙ x = Ax+B(f(x)+
∑
j=j1,···,jp2
gj(x)ρjbjυ0+d),
gj(x)¯ uj
+
∑
j?=j1,···,jp2
y =Cx
(12)
where p2is the total number of the actuators which fail as
(3),
∑
j?=j1,···,jp2
plish the control task. For this condition to be always true,
Assumption 2 is set
Assumption 2: sign(gj(x)) is known.
Without loss of generality, the sign of bjis made the same
as the sign of gj(x), so
∑
j?=j1,···,jp2
Let
¯f(x) = f(x)+
j=j1···jp2
¯ g(x) =
∑
j?=j1···jp2
gj(x)ρjbj?= 0 from Assumption 1 to accom-
gj(x)ρjbj> 0 holds.
∑
gj(x)¯ uj
gj(x)ρjbj
(13)
Then the formation (12) can be described as
˙ x = Ax+B(¯f(x)+ ¯ g(x)υ0+d),
y =Cx
(14)
Assumption 3: Assume x is available.
Then, if¯f(x) and ¯ g(x) are known, and there is no external
disturbance come into the system, the control signal
υi
0=
1
¯ g(x)(−¯f(x)−kTx+r)
(15)
with k = (kn··· k1)Tmakes the system (14) meet that
˙ x = Ax−BkTx+Br = Amx+Bmr
(16)
which exactly matches the reference model if x(0) = xm(0).
Let e = x−xm, then from (4) and (16), one can get
˙ e = Ame
(17)
From (5) we see that the matrix Amis defined stable, so there
exit symmetric positive definite matrices P and Q such that
AT
mP+PAm≤ −Q
(18)
is satisfied. Then from the Lyapnov stability theory, it can
be seen that lim
t→∞e = 0, thus lim
can be drawn that if the nonlinear functions¯f(x) and ¯ g(x)
t→∞y = ym. So, the conclusion
are known, and there is no disturbance, with the control law
(15), the output of system (1) will asymptotically track the
output of the reference model (4).
But the problem is that the nonlinear functions¯f(x) and
¯ g(x) are unknown, thus the ideal control law (15) can not
be applied directly. Since FLS are universal approximators,
they can be used to approximate the unknown functions.
Lemma 1:
For any given real continuous function F(x),
on a compact set U ⊆Rn, there exits an FLS of the following
form that can uniformly approximate F(x) overU to arbitrary
accuracy.
Y(x) = θTξ(x)
where θ =(θ1,θ2,··· ,θM)Tis the estimate parameter vector,
and ξ(x) = (ξ1(x),ξ2(x),··· ,ξM(x))Tis the vector of fuzzy
basis functions, M is the number of fuzzy rules. One can
refer to [16] and [17] for more details.
We make the fuzzy approximation of¯f(x) and ¯ g(x) as
ˆ¯f(x) = θT
ˆ¯ g(x) = θT
fξ(x)
gξ(x)
where estimate parameter vectors θf and θgcan be adjusted
by the corresponding adaptive laws respectively. Define the
optimal parameters as
θ∗
θ∗
f= argmin
g= argmin?sup??ˆ¯ g(x)− ¯ g(x)???
ωe= (ˆ¯f(x|θ∗
= (θ∗
f
?
sup
???ˆ¯f(x)−¯f(x)
???
?
The minimum approximation error of FLS is
f)−¯f(x))+(ˆ¯ g(x|θ∗
Tξ(x)−¯f(x))+(θ∗
g)− ¯ g(x))υ0
Tξ(x)− ¯ g(x))υ0
g
(19)
The estimate parameter errors are
˜θf= θf−θ∗
˜θg= θg−θ∗
f
g
(20)
The estimate of the ideal control law (15) is obtained as
ˆ υi
0=
1
ˆ¯ g(x)(−ˆ¯f(x)−kTx+r) =
1
θT
gξ(x)(−θT
fξ(x)−kTx+r)
(21)
And note that the disturbance should be taken into account
when design the applied control law. Inspired by [19], we
employ extra control signal ua to attenuate the external
disturbance d and the fuzzy logic approximation error ωe
in (19).
ua= −1
rBTPe
(22)
where r is a positive scalar value satisfies the following
Riccati-like equation
AT
mP+PAm+Q−2
rPBBTP+1
η2PBBTP = 0(23)
η is the prescribed attenuation level.Then the applied control
signal is designed as
υ0=
1
θT
gξ(x)(−θT
fξ(x)−kTx+ua+r)
(24)
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We design the adaptive laws as follows:
˙θf= γ1eTPBξ(x)



˙θgi=
γ2eTPBξi(x)υ0
if θgi> δ or
(θgi= δ and eTPBξi(x)υ0> 0)
otherwise
0
(25)
where γ1and γ2are the adaptive gain, and δ is a chosen small
positive constant. The project algorithm of the adaptive law
is used to keep ˆ¯ g(x) off an neighborhood of zero, so that the
proposed control law will be nonsingular. Here we think that
the estimate parameters θf and θgare within certain bound
by the adjustment of (25) to clarify the presentation. The
adaptive fuzzy control scheme has the following properties.
Theorem 1: With the designed controller (24) and the
adaptive laws (25), the controlled unknown nonlinear system
(1) with external disturbance and unknown actuator failures
(2) and/or (3) will achieve the control objective that all
closed-loop signals are bounded, and the output robustly
tracks the output of the reference model (4) with the fol-
lowing H∞performance:
?tj
tj−1eTQe dt ≤ eT(t+
j−1)Pe(t+
γ2˜θT
j−1)+1
j−1)+η2?tj
γ1˜θT
f(t+
j−1)˜θf(t+
tj−1ωTωdt
j−1)+
1
g(t+
j−1)˜θg(t+
(26)
where ω = (ωe−d) ∈ L2(tj−1,tj), (tj−1,tj) is the interval
where the actuator failure pattern is unchanged.
Proof: Take the control (24) into the system formulation
(14), and by the reference model (4), we can get the
following error equation
˙ e = Ax+B(¯f(x)−ˆ¯f(x)+ ¯ g(x)υ0−ˆ¯ g(x)υ0+d−kTx+ua+
r)−Amxm−Bmr
= Ax−BkTx+Br−B[(θf−θ∗
υ0+(θ∗
f
−Amxm−Bmr
= Ame−B(˜θT
gξ(x)υ0−ua−ω)
f)Tξ(x)+(θg−θ∗
Tξ(x)− ¯ g(x))υ0−ua−d]
g)Tξ(x)
Tξ(x)−¯f(x))+(θ∗
g
fξ(x)+˜θT
(27)
Suppose that one or more than one actuators fail at time
instant tj, j = 1, 2, ··· , q, 1 ≤ q ≤ m−1 if all faults are of
the form (3), 1 ≤ q ≤ m otherwise. And at time t ∈ (tj−1,tj),
there are p1(0 ≤ p1≤ m) actuators fail as (2), and p2(0 ≤
p2≤ m−1) actuators fail as (3). Define Lyapunov function
on the interval (tj−1,tj), as
Vj−1=1
2eTPe+
1
2γ1
˜θT
f˜θf+
1
2γ2
˜θT
g˜θg
(28)
It’s derivative along error equation (27) with the control
signal (24) is
˙Vj−1=1
2
2[eTAT
eTPAme−eTPB(˜θT
1
γ1˜θT
f
?˙ eTPe+eTP˙ e?+1
mPe−(˜θT
γ1˜θT
f
˙˜θf+1
gξ(x)υ0−ua−ω)BTPe+
fξ(x)+˜θT
γ2˜θT
g˙˜θg
=1
fξ(x)+˜θT
gξ(x)υ0−ua−ω)]+
˙˜θf+1
γ2˜θT
g˙˜θg
(29)
By the definition of uain (22), (29) can be rewritten as
˙Vj−1=1
2eT(AT
ξ(x)−eTPB˜θT
= −1
= −1
≤ −1
mP+PAm−2
rPBBTP)e+eTPBω −eTPB˜θT
gξ(x)υ0+1
f
2η2eTPBBTPe+eTPBω −ε
2eTQe−1
2eTQe+1
f
γ1˜θT
˙˜θf+1
γ2˜θT
g˙˜θg
2eTQe−
1
2(1
2η2ω2−ε
ηeTPB−ηω)2+1
2η2ω2−ε
(30)
Where the Riccati-like equation (23), the adaptive laws (25)
and the fact that˙˜θf =˙θf,˙˜θg=˙θg are considered. ε is
defined as follows: ε = 0, if for all i, θgi> δ or θgi=
δ and ξTPBζiυ0> 0; ε =∑i˜θgiξTPBζiυ0, otherwise. It can
be proved easily that ε ≥ 0, then the following inequality is
obtained
˙Vj−1≤ −1
2eTQe+1
2η2ω2
(31)
Integrating both sides of (31) from tj−1to tj, and by the de-
finition of the Lyapunov function (28), the H∞performance
(26) can be obtained, which means that the tracking error e
is bounded. Because θf and θg are bounded with adaptive
laws (25), then all closed-loop signals are bounded. Thus
Theorem 1 has been proven.
Remark 2: If θf and θg are bounded with (25), the
following parameter Projection algorithm can be implied to
modify the adaptive laws
˙θf=






γ1eTPBξ(x)
if ?θf? < Mfor
(?θf? = Mfand eTPBθT
if ?θf? = Mfand eTPBθT
if ?θg? < Mgor(?θg? = Mgand
eTPBθT
gξ(x)υ0< 0)
if ?θg? = Mgand eTPBθT
fξ(x) < 0)
fξ(x) > 0
Pf[·]
γ2eTPBξ(x)υ0
˙θg=
Pg[·]
gξ(x)υ0> 0
where Mfand Mgare the boundary of θfand θgrespectively,
and
Pf[·] = γ1eTPBξ(x)−θf
γ1eTPBθT
?θf?2
fξ(x)
Pg[·] = γ2eTPBξ(x)υ0−θg
γ2eTPBθT
gξ(x)υ0
?θg?2
then θf and θgcan be constrained to be bounded.
Remark 3: Actuators fail at time t = tj, j = 1, 2, ··· , q,
then there may be a finite jumping in the optimal parameters
θ∗
the new Lyapunov function Vj on the next time interval
(tj,tj+1). Consequently, the H∞boundary of the output
tracking will jump infinitely. If after some actuator failures,
the the boundary of the output tracking error will jump to
a level that not be satisfying any more, then we can tune
the attenuation level η to further decrease the boundary of
the output tracking error. This can be reasonable because
computer is used to achieve the control task, so the prescribed
attenuation level η can be tuned on-line. For example, if
|e| ≤ meis requested after the initial transient adjustment. If
after some time t, |e| > me, then a higher attenuation level
η will be adopted to recover the requested output tracking
fand θ∗
g, so˜θf and˜θg will change their values to form
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performance if it is possible. So the proposed control scheme
can guarantee the tracking performance as well as the closed-
loop stability of the controlled system with actuator failures.
IV. SIMULATION EXAMPLE
In this section, the presented adaptive fuzzy fault tolerant
control law is applied to a nonlinear system with external
disturbance and actuator faults described as (2) and (3).
Example: We consider the nonlinear system which can be
written as the following form with redundant actuators and
external disturbance.
˙ x1= x2
˙ x2=5sinx1−0.02x2
2cosx1sinx1
3−0.2cosx1cosx2
2cos2x1
3−0.2cosx1cosx2u2+d
+
cos2x1
3−0.2cosx1cosx2u1+
(32)
where the actuator of u1is stuck at 2 at t = 4s, i.e. uF
when t ≥ 4, while the actuator of u2loses 80% effectiveness
at t = 13s that is uF
2= 0.2u2when t ≥ 13 in simulation. The
situation of the failure is already very severe. The external
disturbance d is assumed to be a square wave with the
amplitude 0.05 and the period 2π. The reference model is
as follows:
1= 2
?
˙ xm1
˙ xm2
?
=
?
01
−1
−2
??
xm1
xm2
?
+
?
0
1
?
r
where xm= (xm1, xm2)T= (π/30sin(t), π/30cos(t)), r =
π/15cos(t). Choose k1= 2, k2= 1, and Q = diag(10,10).
Then by solving Riccati-like equation (23) one can obtain
P =
?
15
5
5
5
?
γ1= 0.1, and γ2= 0.01 for adaptive adjusting. We define
seven fuzzy sets over each axis, which label as F1
i = 1,2, respectively. The fuzzy membership functions are
i, ···, F7
i,
µF1
µF2
µF3
µF4
µF5
µF6
µF7
i(xi) =
i(xi) = exp(−(xi+0.4)2)
i(xi) = exp(−(xi+0.2)2)
i(xi) = exp(−x2
i(xi) = exp(−(xi−0.2)2)
i(xi) = exp(−(xi−0.4)2)
i(xi) =
1+exp(−5×(xi−0.6))
1
1+exp(5×(xi+0.6))
i)
1
So we have M =7×7=49 rules for each fuzzy logic system.
The initial values are selected as θf(0)=0, θg(0)=0.2I49×1,
and x(0)=(0.2, 0.2)T. Two cases are simulated: in one case,
η is 0.1 for all the time; while in the other case, η is changed
from 0.1 to 0.05 when the condition 100?t
which is equivalent to |e| > me is met due to the severe
actuator failures. The simulation results of the both cases for
0 ≤t ≤ 40 are shown in Fig.1 - Fig.4
We can see from the results that with the proposed control
scheme, the controlled system can be stable and achieve
robust output tracking performance. However, some severe
actuator failures may cause obvious tracking error between
the output of the system and the reference model (see Fig.1
0eTe dt > 2.1
0510 15 20 2530 3540
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
time(s)
Position tracking
Fig. 1.The output tracking with unvaried η
0510152025303540
0
0.5
1
1.5
2
2.5
time(s)
integration of e2
Fig. 2.100?t
0eTe dt with unvaried η
0510152025303540
−0.15
−0.1
−0.05
0
0.05
0.1
0.15
0.2
0.25
time(s)
Position tracking
Fig. 3.The output tracking with variable η
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05101520 25303540
0
0.5
1
1.5
2
2.5
time(s)
∫ eTQedt
Fig. 4.100?t
0eTe dt with variable η
and Fig.2). If the attenuation level can be changed higher on-
line when the output tracking performance is not satisfying,
the tracking error can be decreased effectively to get almost
perfect output tracking (see Fig.3 and Fig.4).
V. CONCLUSION
This paper studies fault tolerant control problem for un-
known nonlinear systems with actuator failures and external
disturbance. With the developed adaptive fuzzy control law,
the closed-loop system can be guaranteed stable and the
output tracks the reference model output robustly. This is
achieved by an adaptive control law with fuzzy logic systems
approximating the unknown system functions and actuator
failures together, and attenuation technique to attenuate the
influence of both fuzzy logic approximation error and exter-
nal disturbance on the tracking error to a prescribed level,
the adaptive laws adjusting estimate parameters in the fuzzy
logic approximators are obtained on the basis of Lyapunov
stability theory. Note that the proposed adaptive fuzzy fault
tolerant control approach need not resort to the fault detection
and isolation mechanism by adaptively adjust the control
law, thus the undesired system behavior caused by false
alarm or omitted alarm can be avoided. And the fault under
consideration can be lock-in-place, loss of effectiveness or
both of them, so the types of actuator fault that can be
tolerant have been broadened even in unknown nonlinear
system with external disturbance.The simulation results show
the effectiveness of the proposed control scheme though the
fault occurred is severe.
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