# Adaptive Robust Actuator Fault Accommodation for a Class of Uncertain Nonlinear Systems with Unknown Control Gains

Authors: Yuefei Wu; Dawei Ma; Jianyong Yao; Guigao Le; Zhenxing Mu; Fan Yang
Journal: Mathematical Problems in Engineering
 (2014)

---

## Abstract

An adaptive robust fault tolerant control approach is proposed for a class of uncertain nonlinear systems with unknown signs of high-frequency gain and unmeasured states. In the recursive design, neural networks are employed to approximate the unknown nonlinear functions, K-filters are designed to estimate the unmeasured states, and a dynamical signal and Nussbaum gain functions are introduced to handle the unknown sign of the virtual control direction. By incorporating the switching functionσ algorithm, the adaptive backstepping scheme developed in this paper does not require the real value of the actuator failure. It is mathematically proved that the proposed adaptive robust fault tolerant control approach can guarantee that all the signals of the closed-loop system are bounded, and the output converges to a small neighborhood of the origin. The effectiveness of the proposed approach is illustrated by the simulation examples.


---

## Body

## 1. Introduction

In complex systems like chemical plants, nuclear reactors, and flight control systems, reliability is as important as performance. Conventional feedback design [1] for such complex systems may result in unacceptable degradation in performance or even instability in the event such as actuators, sensors, and processors that may undergo abrupt failures individually or simultaneously during operation. The adverse effects due to the failures require being compensated to enhance the reliability and safety of the system. The research on accommodating such failures and maintaining acceptable system performance is particularly important. System faults are typically characterized by critical changes in the system parameters and changes in the inherent dynamical structure of the system. Hence, effective fault diagnosis and accommodation (FDA) have become an important area of research [2–4].In this work, we focus on the problem of actuator failure accommodation. Various approaches to FDA using analytical redundancy have been reported during the last three decades. Generally speaking, the control methods can be clarified into the following types: fault detection and diagnosis designs [5–7]; linear matrix inequality techniques [8, 9]; adaptive approaches [10, 11]; and so forth. Among these design methods, adaptive mechanisms [12–16] have been employed, and adaptive control has been a promising approach to deal with such failures. In adaptive control systems, controllers were designed with the aid of adaptation mechanisms to handle large uncertain structural and parametric variation caused by failures. In [17], a novel attempt was made to compensate for the actuator failures in linear time-invariant systems by using adaptive state feedback. However, all states were assumed to be available for this proposed scheme. As noted in [18], in practice, state variables were often immeasurable for many practical nonlinear systems. In such cases, an adaptive output-feedback control scheme should be developed. In [19], an adaptive output-feedback controller was synthesized. Furthermore, nonlinear systems with actuator failures were investigated. In [20], adaptive state feedback failure compensation schemes were proposed for nonlinear systems in the parametric strict-feedback form. However, nonlinear behaviors [10, 21–23] and modeling uncertainties complicate the development of high-performance closed-loop controllers. A robust adaptive state feedback failure compensation method considering modeling uncertainties was proposed in [12, 24, 25]. Apparently, such techniques were not well suited to suppress the undesirable transients when facing a sudden change in system parameters due to unknown actuator faults. In [26, 27], a robust model-based fault detection scheme was developed by using adaptive robust strategy to deal with parametric uncertainties and bounded uncertainty nonlinearities.Recently, the problem of adaptive control of systems with the unknown sign of high frequency gain has also received much attention. How to weaken the high frequency gain sign assumptions is an important issue. The Nussbaum-type function was originally proposed by [28] for dealing with unknown sign of high frequency gain. This method was then generalized to higher order linear systems by [29]. For a class of time-varying parameter high-order uncertain nonlinear systems, a robust adaptive output-feedback control method was proposed in [30–33] for the unknown control gain direction and unpredictable state. However, the proposed approaches were only focused on the so-called nonlinear strict-feedback systems, in which the nonlinear uncertainties were known or can be linearly parameterized. In [34], an adaptive neural network backstepping control scheme has been developed, an adaptive neural network backstepping control scheme for a given class of nonlinear systems, and neural network systems were used to approximate the unknown nonlinear functions, and the stability of the closed-loop system was given based on iterative Lyapunov design. This result has been extended by [35] to a class of the nonlinear time-delay systems in the strict-feedback form. However, there is little work using this method to deal with unknown actuator failures and unknown control gain simultaneously.In this paper, we propose an adaptive robust approach for actuator fault-tolerant control (ARFTC) of a class of uncertain nonlinear systems. Moreover, compared with [10, 12, 24], a parameterisable time-varying actuator failure model is investigated. The technique here is a combination of adaptive backstepping [36] and switching function σ algorithm based ARFTC proposed in [27] and differs significantly from the techniques presented in [19] which relies on backstepping based direct adaptive control. Specifically, ARFTC uses robust filter structures to attenuate the effect of model uncertainties, and adaptation is used only as a means to reduce the extent of parametric uncertainties. However, neither adaptive control nor robust control based fault-tolerant designs can address the issues associated with actuator faults. In the present work, we claim that an adaptive robust fault-tolerant control scheme integrates adaptive and robust control design techniques. In order to show the superior performance of the proposed scheme, comparative studies are performed using simulation examples.


## 2. Problem Formulation and Preliminaries

### 2.1. Problem Formulation

In this paper, we consider the following nonlinear system as [27]
(1)
i
i
l
x
˙
1
=
x
2
+
f
1
(
y
)
+
φ
0,1
(
y
)
+
d
1
,
aaaaa
⋮
x
˙
ρ
-
1
=
x
ρ
+
f
ρ
-
1
(
y
)
+
φ
0
,
ρ
-
1
(
y
)
+
d
ρ
-
1
(
t
,
x
)
,
i
i
i
x
˙
ρ
=
x
ρ
+
1
+
f
ρ
(
y
)
+
φ
0
,
ρ
(
y
)
+
∑
j
=
2
ρ
f
j
,
ρ
(
y
)
x
j
aaaaaa
+
∑
j
=
1
q
b
m
,
j
β
j
(
y
)
u
j
+
d
ρ
(
t
,
x
)
,
2
≤
ρ
≤
n
-
1
,
aaaaa
⋮
x
˙
n
-
1
=
x
n
-
1
+
f
n
-
1
(
y
)
+
φ
0
,
n
-
1
(
y
)
+
∑
j
=
2
n
-
1
f
j
,
n
-
1
(
y
)
x
j
aaaaaa
+
∑
j
=
1
q
b
1
,
j
β
j
(
y
)
u
j
+
d
n
-
1
(
t
,
x
)
,
i
i
i
x
˙
n
=
φ
0
,
n
(
y
)
+
f
n
(
y
)
+
∑
j
=
1
q
b
0
,
j
β
j
(
y
)
u
j
aaaaaa
+
∑
j
=
2
n
f
j
,
n
(
y
)
x
j
+
d
n
(
t
,
x
)
,
y
=
x
1
,
where u
j
∈
R, j
=
1,2
,
…
,
q are the control inputs whose actuators may fail during operation; x
=
[
x
1
,
x
2
,
…
,
x
n
]
T
∈
R
n is the state vector; y
∈
R is the system output; f
i
(
y
), i
=
1
,
2
,
…
,
n are unknown smooth functions, which represent model uncertainties due to modeling errors or unmodeled dynamics; d
i
(
t
) are bounded time-varying disturbances with unknown constant bounds; b
i
,
j, i
=
0,1
,
…
,
m, j
=
1
,
2
,
…
,
q are unknown constant parameters; φ
0
,
i
(
y
), i
=
1,2
,
…
,
n, β
j
(
y
), j
=
1
,
2
,
…
,
q are known smooth nonlinear functions; b
i
,
j
β
j
(
y
)
≠
0 for y
∈
R. Only the output y is available for measurement.Assumption 1 (see [27]).
System (1) is such that the desired control objective can be fulfilled with up to m
-
1 stuck actuators, the remaining actuators can still achieve a desired control objective when implemented with the knowledge of the plant parameters and failure parameters.Assumption 2 (see [26]).
The relative degreeρ
=
n
-
m is known and the system is minimum phase; the polynomial B
(
s
)
=
b
m
s
m
+
⋯
+
b
1
s
+
b
0 is Hurwitz.Assumption 3.
The unknown disturbanced
i
(
t
) satisfies |
d
i
(
t
)
|
≤
d
i, where d
i is an unknown constant.A time-varying actuator failure can be modeled as [27]
(2)
u
j
(
t
)
=
u
-
j
+
d
-
j
(
t
)
,
t
≥
t
j
,
j
∈
{
1,2
,
…
,
q
}
,
where the failure value u
-
j, the failure time instant t
j, and the failure index j are unknown; d
-
j
(
t
) is given by
(3)
d
-
j
(
t
)
=
∑
l
=
1
h
d
-
j
l
f
j
l
(
t
)
for some unknown scalar constants d
-
j
l and known bounded scalar signals f
j
l
(
t
), j
=
1,2
,
…
,
q, l
=
1,2
,
…
,
h, h
≥
1.Thus, the actuator failure mode is defined as [26]
(4)
u
j
(
t
)
=
{
u
-
j
+
d
-
j
(
t
)
,
if stuck actuator,
t
≥
T
f
η
j
ν
j
(
t
)
,
if loss of efficiency,
t
≥
T
f
,
where ν
j
(
t
), j
=
1,2
,
…
,
q are applied control signals from a feedback control design, and where T
f is the unknown instant of failure, u
-
j is an unknown constant value at which the actuator gets stuck, and η
j
∈
[
(
η
j
)
min
⁡
,
1
] represents actuator loss in efficiency.The control target is that all the closed-loop signals remain bounded, while the plant outputy
(
t
) asymptotically tracks a prescribed signal y
d
(
t
) despite the presence of unknown actuator failures, unknown plant parameters, and unknown control gain signs. The reference signal y
d
(
t
) and its derivatives are known and bounded.


### 2.2. Nussbaum Function Properties

To deal with the unknown control gain signs, we introduce the knowledge of Nussbaum-type gain. A smooth functionN
(
k
)
:
R
→
R is called Nussbaum-type gain if it has the following properties [29]
(5)
lim
⁡
s
→
+
∞
sup
⁡
1
s
∫
0
s
N
(
ς
)
d
ς
=
+
∞
,
lim
⁡
s
→
+
∞
inf
⁡
1
s
∫
0
s
N
(
ς
)
d
ς
=
-
∞
.For instance,k
2
cos
⁡
(
k
) and k
2
sin
(
k
) belong to this class of functions. In this paper, an even Nussbaum-type function k
2
cos
⁡
(
k
) is used.Lemma 4 (see [35]).
LetV
(
·
) and k
(
·
) be smooth functions defined on [
0
,
t
f
) with V
(
t
)
≥
0, t
∈
[
0
,
t
f
), and N
(
·
) are an even smooth Nussbaum-type function. If the following inequality holds
(6)
V
(
t
)
≤
c
0
+
e
-
c
1
τ
∫
0
t
[
x
(
τ
)
N
(
k
)
+
1
]
k
˙
e
c
1
τ
d
τ
,
where c
0 represents a suitable constant, c
1 is a positive constant, and x
(
t
) is a time-varying parameter taking values in the unknown closed intervals, then V
(
t
), k
(
t
), ∫
0
t
x
(
τ
)
N
(
k
)
k
˙
d
τ must be bounded on [
0
,
t
f
).


### 2.3. Neural Networks (NNs)

NNs have been widely used in modeling and control of nonlinear systems due to their good capabilities of nonlinear function approximation, learning, and fault tolerance [34]. The following radial basis function NNs (RBFNNs) are used to approximate the continuous function f
i
(
y
)
:
R
→
R:
(7)
f
i
(
y
)
=
W
i
T
𝚽
i
(
y
)
,
where the input y
∈
R; the weight vector Φ
i
=
[
Φ
i
1
,
…
,
Φ
i
l
]
T, W
i
=
[
ω
i
1
,
…
,
ω
i
l
]
T with the NN node number l; the vector of smooth basis functions Φ
i
j being chosen as the commonly used Gaussian functions 𝚽
i
=
exp
⁡
(
-
(
y
-
c
i
)
2
/
(
η
i
2
)
)
,
i
=
1,2
,
…
,
l, where c
i
=
[
c
i
1
,
…
,
c
i
l
]
T is the center of the receptive field and η
i is the width of the Gaussian function. It has been proven in [34] that networks (7) can approximate any smooth functions over a compact set y
∈
R to accuracy as
(8)
f
i
(
y
)
=
W
i
*
T
𝚽
i
(
y
)
+
ε
i
,
where W
i
* is ideal constant weights, and the approximation error ε
i
(
y
) satisfies |
ε
i
(
y
)
|
≤
ε
i with constant ε
i
>
0.The ideal weight vectorW
i
* is defined as the value of W
i
* that minimizes |
ε
i
(
y
)
|; that is,
(9)
W
i
*
=
arg
min
⁡
W
i
{
sup
⁡
y
∈
Ω
y
|
f
i
(
y
)
-
W
^
i
T
𝚽
i
(
y
)
|
}
.


## 2.1. Problem Formulation

In this paper, we consider the following nonlinear system as [27]
(1)
i
i
l
x
˙
1
=
x
2
+
f
1
(
y
)
+
φ
0,1
(
y
)
+
d
1
,
aaaaa
⋮
x
˙
ρ
-
1
=
x
ρ
+
f
ρ
-
1
(
y
)
+
φ
0
,
ρ
-
1
(
y
)
+
d
ρ
-
1
(
t
,
x
)
,
i
i
i
x
˙
ρ
=
x
ρ
+
1
+
f
ρ
(
y
)
+
φ
0
,
ρ
(
y
)
+
∑
j
=
2
ρ
f
j
,
ρ
(
y
)
x
j
aaaaaa
+
∑
j
=
1
q
b
m
,
j
β
j
(
y
)
u
j
+
d
ρ
(
t
,
x
)
,
2
≤
ρ
≤
n
-
1
,
aaaaa
⋮
x
˙
n
-
1
=
x
n
-
1
+
f
n
-
1
(
y
)
+
φ
0
,
n
-
1
(
y
)
+
∑
j
=
2
n
-
1
f
j
,
n
-
1
(
y
)
x
j
aaaaaa
+
∑
j
=
1
q
b
1
,
j
β
j
(
y
)
u
j
+
d
n
-
1
(
t
,
x
)
,
i
i
i
x
˙
n
=
φ
0
,
n
(
y
)
+
f
n
(
y
)
+
∑
j
=
1
q
b
0
,
j
β
j
(
y
)
u
j
aaaaaa
+
∑
j
=
2
n
f
j
,
n
(
y
)
x
j
+
d
n
(
t
,
x
)
,
y
=
x
1
,
where u
j
∈
R, j
=
1,2
,
…
,
q are the control inputs whose actuators may fail during operation; x
=
[
x
1
,
x
2
,
…
,
x
n
]
T
∈
R
n is the state vector; y
∈
R is the system output; f
i
(
y
), i
=
1
,
2
,
…
,
n are unknown smooth functions, which represent model uncertainties due to modeling errors or unmodeled dynamics; d
i
(
t
) are bounded time-varying disturbances with unknown constant bounds; b
i
,
j, i
=
0,1
,
…
,
m, j
=
1
,
2
,
…
,
q are unknown constant parameters; φ
0
,
i
(
y
), i
=
1,2
,
…
,
n, β
j
(
y
), j
=
1
,
2
,
…
,
q are known smooth nonlinear functions; b
i
,
j
β
j
(
y
)
≠
0 for y
∈
R. Only the output y is available for measurement.Assumption 1 (see [27]).
System (1) is such that the desired control objective can be fulfilled with up to m
-
1 stuck actuators, the remaining actuators can still achieve a desired control objective when implemented with the knowledge of the plant parameters and failure parameters.Assumption 2 (see [26]).
The relative degreeρ
=
n
-
m is known and the system is minimum phase; the polynomial B
(
s
)
=
b
m
s
m
+
⋯
+
b
1
s
+
b
0 is Hurwitz.Assumption 3.
The unknown disturbanced
i
(
t
) satisfies |
d
i
(
t
)
|
≤
d
i, where d
i is an unknown constant.A time-varying actuator failure can be modeled as [27]
(2)
u
j
(
t
)
=
u
-
j
+
d
-
j
(
t
)
,
t
≥
t
j
,
j
∈
{
1,2
,
…
,
q
}
,
where the failure value u
-
j, the failure time instant t
j, and the failure index j are unknown; d
-
j
(
t
) is given by
(3)
d
-
j
(
t
)
=
∑
l
=
1
h
d
-
j
l
f
j
l
(
t
)
for some unknown scalar constants d
-
j
l and known bounded scalar signals f
j
l
(
t
), j
=
1,2
,
…
,
q, l
=
1,2
,
…
,
h, h
≥
1.Thus, the actuator failure mode is defined as [26]
(4)
u
j
(
t
)
=
{
u
-
j
+
d
-
j
(
t
)
,
if stuck actuator,
t
≥
T
f
η
j
ν
j
(
t
)
,
if loss of efficiency,
t
≥
T
f
,
where ν
j
(
t
), j
=
1,2
,
…
,
q are applied control signals from a feedback control design, and where T
f is the unknown instant of failure, u
-
j is an unknown constant value at which the actuator gets stuck, and η
j
∈
[
(
η
j
)
min
⁡
,
1
] represents actuator loss in efficiency.The control target is that all the closed-loop signals remain bounded, while the plant outputy
(
t
) asymptotically tracks a prescribed signal y
d
(
t
) despite the presence of unknown actuator failures, unknown plant parameters, and unknown control gain signs. The reference signal y
d
(
t
) and its derivatives are known and bounded.


## 2.2. Nussbaum Function Properties

To deal with the unknown control gain signs, we introduce the knowledge of Nussbaum-type gain. A smooth functionN
(
k
)
:
R
→
R is called Nussbaum-type gain if it has the following properties [29]
(5)
lim
⁡
s
→
+
∞
sup
⁡
1
s
∫
0
s
N
(
ς
)
d
ς
=
+
∞
,
lim
⁡
s
→
+
∞
inf
⁡
1
s
∫
0
s
N
(
ς
)
d
ς
=
-
∞
.For instance,k
2
cos
⁡
(
k
) and k
2
sin
(
k
) belong to this class of functions. In this paper, an even Nussbaum-type function k
2
cos
⁡
(
k
) is used.Lemma 4 (see [35]).
LetV
(
·
) and k
(
·
) be smooth functions defined on [
0
,
t
f
) with V
(
t
)
≥
0, t
∈
[
0
,
t
f
), and N
(
·
) are an even smooth Nussbaum-type function. If the following inequality holds
(6)
V
(
t
)
≤
c
0
+
e
-
c
1
τ
∫
0
t
[
x
(
τ
)
N
(
k
)
+
1
]
k
˙
e
c
1
τ
d
τ
,
where c
0 represents a suitable constant, c
1 is a positive constant, and x
(
t
) is a time-varying parameter taking values in the unknown closed intervals, then V
(
t
), k
(
t
), ∫
0
t
x
(
τ
)
N
(
k
)
k
˙
d
τ must be bounded on [
0
,
t
f
).


## 2.3. Neural Networks (NNs)

NNs have been widely used in modeling and control of nonlinear systems due to their good capabilities of nonlinear function approximation, learning, and fault tolerance [34]. The following radial basis function NNs (RBFNNs) are used to approximate the continuous function f
i
(
y
)
:
R
→
R:
(7)
f
i
(
y
)
=
W
i
T
𝚽
i
(
y
)
,
where the input y
∈
R; the weight vector Φ
i
=
[
Φ
i
1
,
…
,
Φ
i
l
]
T, W
i
=
[
ω
i
1
,
…
,
ω
i
l
]
T with the NN node number l; the vector of smooth basis functions Φ
i
j being chosen as the commonly used Gaussian functions 𝚽
i
=
exp
⁡
(
-
(
y
-
c
i
)
2
/
(
η
i
2
)
)
,
i
=
1,2
,
…
,
l, where c
i
=
[
c
i
1
,
…
,
c
i
l
]
T is the center of the receptive field and η
i is the width of the Gaussian function. It has been proven in [34] that networks (7) can approximate any smooth functions over a compact set y
∈
R to accuracy as
(8)
f
i
(
y
)
=
W
i
*
T
𝚽
i
(
y
)
+
ε
i
,
where W
i
* is ideal constant weights, and the approximation error ε
i
(
y
) satisfies |
ε
i
(
y
)
|
≤
ε
i with constant ε
i
>
0.The ideal weight vectorW
i
* is defined as the value of W
i
* that minimizes |
ε
i
(
y
)
|; that is,
(9)
W
i
*
=
arg
min
⁡
W
i
{
sup
⁡
y
∈
Ω
y
|
f
i
(
y
)
-
W
^
i
T
𝚽
i
(
y
)
|
}
.


## 3. Output-Feedback Based ARFTC

### 3.1. State Estimation

Control signalsu
j
(
t
) are designed such that u
*
=
u
j
β
(
y
). With fault model (2) and the chosen actuation scheme, we can rewrite the control inputs as follows:
(10)
u
j
(
t
)
=
η
j
β
j
(
y
)
(
1
-
σ
j
)
u
*
(
t
)
+
σ
j
u
-
j
,
j
=
1
,
…
,
q
,
where σ
j
=
1 corresponds to stuck actuator, and σ
j
=
0, η
j
∈
[
(
η
j
)
min
⁡
,
1
] represents efficiency loss of the actuator. In accordance with this, we rewrite the system as follows:
(11)
x
˙
=
A
x
+
φ
0
(
y
)
+
W
*
T
𝚽
(
y
)
+
F
(
y
)
x
+
∑
i
=
0
m
e
n
-
i
κ
i
u
*
+
∑
j
=
1
q
∑
i
=
0
m
e
n
-
i
μ
i
,
j
β
j
(
y
)
+
Δ
(
y
,
t
)
,
y
=
c
T
x
,
where e
1 denotes the first coordinate vector in R
n
(12)
A
=
[
0
⋮
I
n
-
1
0
0
]
,
Φ
(
y
)
=
[
Φ
1
(
y
)
0
⋯
0
0
Φ
2
(
y
)
⋱
⋮
⋮
⋮
⋱
0
0
0
⋯
Φ
n
(
y
)
]
,
F
(
y
)
=
[
0
0
⋯
0
0
f
2,2
⋱
⋮
⋮
⋮
⋱
0
0
f
2,2
⋯
f
n
,
n
]
,
φ
0
(
y
)
=
[
φ
0,1
(
y
)
φ
0,2
(
y
)
⋮
φ
0
,
n
(
y
)
]
,
Δ
=
[
d
1
+
ε
1
d
2
+
ε
2
⋮
d
n
+
ε
n
]
,
κ
i
=
∑
j
=
1
q
η
j
(
1
-
σ
j
)
b
i
,
j
,
μ
i
,
j
=
σ
j
u
-
j
b
i
,
j
,
c
=
[
1
0
⋯
0
]
T
,
W
*
=
[
W
1
*
W
2
*
⋯
W
n
*
]
,
where i
=
0
,
1
,
…
,
m, j
=
1
,
2
,
…
,
q. It can be deduced from Assumption 3 that |
Δ
(
x
,
t
)
|
≤
ε
i
+
d
i
<
ψ
m, where ψ
m is an unknown bounded constant.Note thatκ
i is the unknown measure of actuator effectiveness after faults and μ
i
,
j is the unknown measure of the fault magnitude which needs to be compensated.Thus, the states of system (1), unknown constants and parameter vectors, should be estimated by using the filters given in [26, 27]. We will define the following set of filters for the purpose of state-estimation:
(13)
ξ
˙
0
=
(
A
-
l
L
q
c
T
)
ξ
0
+
l
L
q
y
+
φ
0
(
y
)
+
F
(
y
)
ξ
0
,
ξ
0
∈
R
n
×
1
ξ
˙
=
(
A
-
l
L
q
c
T
)
ξ
+
Φ
(
y
)
,
ξ
∈
R
n
×
l
n
ν
˙
i
=
(
A
-
l
L
q
c
T
)
ν
i
+
e
n
-
i
u
*
+
F
(
y
)
ν
i
,
ν
i
∈
R
n
×
1
ψ
˙
i
,
j
=
(
A
-
l
L
q
c
T
)
ψ
i
,
j
+
e
n
-
i
β
j
(
y
)
,
ψ
i
,
j
∈
R
n
×
1
,
where the gain matrix q
=
[
q
1
,
q
2
,
…
,
q
n
]
T is chosen to make A
-
q
c
T Hurwitz, i
=
0
,
1
,
…
,
m, j
=
1
,
2
,
…
,
q, L
=
diag
⁡
[
1
l
⋯
l
n
-
1
], and l is the observer gain updated by
(14)
l
˙
=
-
κ
l
2
+
κ
l
+
l
γ
(
y
)
,
l
(
0
)
=
1
with κ a positive design parameter and γ
(
y
) a nonnegative smooth function. It can be proved by contradiction that l
(
t
)
≥
1 for all t
≥
0.Due to the special structure ofA, the order of K-filters can be reduced by using the following two filters:
(15)
λ
˙
=
(
A
-
l
L
q
c
T
)
λ
+
e
n
u
*
+
F
(
y
)
λ
,
ς
˙
j
=
(
A
-
l
L
q
c
T
)
ς
j
+
e
n
β
j
(
y
)
,
j
=
1,2
,
…
,
q
and the following algebraic equations:
(16)
ν
i
=
(
A
-
l
L
q
c
T
)
i
λ
,
ψ
i
,
j
=
(
A
-
l
L
q
c
T
)
i
ς
j
,
i
=
0,1
,
…
,
m
.The estimated state can be written as(17)
x
^
=
ξ
0
+
ξ
W
*
+
∑
i
=
0
m
κ
i
ν
i
+
∑
j
=
1
q
∑
i
=
0
m
μ
i
,
j
ψ
i
,
j
.Letx
~
=
x
-
x
^ be the estimation error. Then, the state estimation error dynamic is given by
(18)
x
~
˙
(
t
)
=
(
A
-
l
L
q
c
T
)
x
~
+
F
(
y
)
x
~
+
Δ
.Noting that the change of coordinatesx
-
=
l
-
μ
L
-
1
x
~ with  μ a positive design parameter (18) is transformed into
(19)
x
-
˙
=
l
(
A
-
q
c
T
)
x
-
+
L
-
1
F
(
y
)
L
x
-
+
l
-
μ
L
-
1
Δ
-
l
˙
l
(
μ
I
+
D
)
x
-
,
where D
=
diag
⁡
{
0
,
1
,
…
,
n
-
1
}. Since A
-
q
c
T is Hurwitz, there is a symmetric positive definite matrix P satisfying P
(
A
-
q
c
T
)
+
(
A
-
q
c
T
)
T
P
=
-
I. Let the quadratic Lyapunov function V
x
=
x
-
T
P
x
-, whose derivative is computed as
(20)
V
˙
x
=
-
l
∥
x
-
∥
2
+
2
x
-
T
P
L
-
1
F
(
y
)
L
x
-
+
2
l
-
μ
x
-
T
P
L
-
1
Δ
-
l
˙
l
x
-
T
(
P
D
+
D
P
+
2
μ
P
)
x
-
.Note thatl
≥
1. Then there is a nonnegative smooth function γ
1
(
y
) such that L
-
1
F
(
y
)
L
≤
γ
1
(
y
), from which it follows that
(21)
2
x
-
T
P
L
-
1
F
(
y
)
L
x
-
≤
2
∥
P
∥
γ
1
(
y
)
∥
x
-
∥
2
.From Young’s inequality, we have(22)
2
l
-
μ
x
-
T
P
L
-
1
Δ
≤
l
-
2
μ
x
-
T
x
-
+
∥
P
L
-
1
∥
2
ψ
m
2
.SinceP is a symmetric positive definite matrix, by choosing a sufficiently large μ, we can obtain
(23)
σ
1
I
≤
D
P
+
P
D
+
2
μ
P
≤
σ
2
I
which together with (14) implies that
(24)
-
l
˙
l
x
-
T
(
P
D
+
D
P
+
2
μ
P
)
x
-
≤
κ
σ
2
l
∥
x
-
∥
2
-
σ
1
γ
(
y
)
∥
x
-
∥
2
,
where σ
1, σ
2 are positive constants. From (21) and (24), (18) can be rewritten as
(25)
V
˙
x
≤
-
[
(
1
-
l
-
2
μ
-
κ
σ
2
)
l
+
σ
1
γ
(
y
)
aaaaaa
-
2
∥
P
∥
γ
1
(
y
)
l
-
2
μ
]
∥
x
-
∥
2
+
∥
P
L
-
1
∥
2
ψ
m
2
.
By choosing κ and γ
(
y
) to satisfy κ
≤
(
1
-
2
l
-
2
μ
)
/
(
2
σ
2
), γ
(
y
)
≥
2
∥
P
∥
γ
1
(
y
)
/
σ
1
≥
0, we arrive at
(26)
V
˙
x
≤
-
1
2
l
∥
x
-
∥
2
+
∥
P
L
-
1
∥
2
ψ
m
2
.


### 3.2. Parameter Estimate

Letθ
^ denote the estimate of θ and θ
~
=
θ
-
θ
^ denote the estimation error. The extent of parametric uncertainties satisfy
(27)
Ω
=
{
θ
∣
∥
θ
∥
≤
M
1
}
,
where M
1 is a positive design parameter.It is well known that parameter estimation algorithms suffer from parameter drift in presence of disturbances, resulting in system states growing unboundedly. We use the switching functionσ algorithm [36] to deal with this problem. The update law used here has the following form:
(28)
σ
θ
(
∥
θ
^
∥
)
=
{
0
,
∥
θ
^
∥
≤
M
1
χ
1
(
∥
θ
^
∥
)
,
M
1
≤
∥
θ
^
∥
≤
2
M
1
σ
10
,
∥
θ
^
∥
≥
2
M
1
,
where σ
10 is a positive design parameter, and χ
1 is an arbitrary adaptation function.Consider(29)
χ
1
(
∥
θ
^
∥
)
=
σ
10
(
2
ρ
-
1
)
!
M
1
2
ρ
-
1
[
(
ρ
-
1
)
!
]
2
∫
M
1
∥
θ
^
∥
(
Θ
-
M
1
)
(
2
M
1
-
Θ
)
d
Θ
.The mapping guarantees that the following properties are always satisfied:(30)
σ
θ
θ
~
T
θ
^
≤
-
1
2
σ
10
θ
~
T
θ
~
+
13
2
σ
10
M
1
2
.


### 3.3. Controller Design

Furthermore, system (13) can be represented as
(31)
ν
˙
m
,
i
=
ν
˙
m
,
i
+
1
-
q
i
l
i
ν
˙
m
,
1
,
i
=
2,3
,
…
,
ρ
-
1
,
ν
˙
m
,
ρ
=
ν
˙
m
,
ρ
+
1
-
q
ρ
l
ρ
ν
˙
m
,
1
+
u
*
.The derivative of the outputy is given by
(32)
y
˙
=
ω
0
+
ω
T
θ
+
Δ
1
(
y
,
t
)
=
κ
m
ν
m
,
2
+
ω
0
+
ω
-
T
θ
+
Δ
1
(
y
,
t
)
,
where the regressor ω and truncated regressor ω
0 are defined as [26]
(33)
ω
0
=
[
ξ
0,2
+
φ
0,1
]
,
ω
=
[
ψ
0
,
q
(
2
)
,
ξ
(
2
)
+
𝚽
(
1
)
,
ν
m
,
2
,
ν
m
-
1,2
,
…
,
ν
0,2
,
a
a
i
a
ψ
m
,
1
(
2
)
,
…
,
ψ
m
,
q
(
2
)
,
…
,
ψ
0,1
(
2
)
,
…
,
ψ
0
,
q
(
2
)
,
]
T
,
ω
-
=
ω
-
e
l
n
+
1
ν
m
,
2
θ
=
[
W
*
,
κ
m
,
…
,
κ
0
,
μ
m
,
1
,
…
,
μ
m
,
q
,
…
,
μ
0,1
,
…
,
μ
0
,
q
]
T
.In this section, we present the adaptive output-feedback control design using the backstepping technique. Define the following error coordinates:z
1
=
y
-
y
d and z
i
=
v
m
,
i
-
α
i
-
1, i
=
2,3
,
…
,
ρ, where α
i
-
1 is the stabilizing functions to be designed.Step 1.
Differentiatingz
1 with respect to time t, we obtain
(34)
z
˙
1
=
κ
m
ν
m
,
2
+
ω
0
+
ω
-
T
θ
-
y
˙
d
+
Δ
1
(
y
,
t
)
.
The problem of the unknown sign of the virtual direction is sloved by the Nussbaum-type functionsκ
m. Choose the tuning functions and parameter adaptation law as
(35)
α
1
=
N
(
k
)
τ
,
k
˙
=
z
1
τ
,
τ
=
c
1
z
1
+
ω
-
T
θ
^
-
y
˙
d
+
l
1
+
2
μ
z
1
+
ψ
^
m
tanh
⁡
(
z
1
δ
1
)
,
where N
(
k
) is Nussbaum gain; ψ
^
m is an estimate of ψ
m with the estimation error ψ
~
m
=
ψ
m
-
ψ
^
m; c
1 is a positive constant. Using the inequality
(36)
0
≤
|
z
1
|
-
z
1
tanh
(
z
1
δ
)
≤
0.2785
δ
,
where δ is a positive design parameter and substituting (35) and (36) into (34) yield
(37)
z
˙
1
=
κ
m
[
z
2
+
N
(
k
)
τ
]
+
Δ
1
(
y
,
t
)
+
θ
~
T
Γ
(
τ
1
-
θ
^
˙
)
+
σ
θ
θ
~
T
θ
^
+
τ
-
c
1
z
1
-
l
1
+
2
μ
z
1
-
ψ
^
m
tanh
(
z
1
δ
1
)
,
where τ
1
=
z
1
Γ
-
1
ω
-
-
σ
θ
(
∥
θ
^
∥
)
Γ
-
1
θ
^.
Define the quadratic function(38)
V
1
=
1
2
z
1
2
+
1
2
θ
~
T
Γ
θ
~
+
1
2
r
ψ
ψ
~
m
2
.
From Young’s inequality, we obtain(39)
z
1
x
~
2
≤
l
1
+
2
μ
z
1
2
+
l
x
-
2
2
4
.
DifferentiatingV
1 with respect to time t leads to
(40)
V
˙
1
=
κ
m
z
1
z
2
-
c
1
z
1
2
+
κ
m
z
1
N
(
k
)
τ
+
k
˙
+
1
4
l
x
-
2
2
+
θ
~
T
Γ
(
τ
1
-
θ
^
˙
)
+
σ
θ
θ
~
T
θ
^
+
σ
ψ
ψ
~
m
T
ψ
^
m
+
1
r
ψ
ψ
~
m
(
r
ψ
τ
1
ψ
-
ψ
^
˙
m
)
+
0.2785
δ
1
ψ
m
,
where τ
1
ψ
=
z
1
tanh
(
z
1
/
δ
1
)
-
σ
ψ
ψ
^
m.Step 2.
The time derivative ofz
2 along with (31) is
(41)
z
˙
2
=
ν
m
,
3
-
q
2
l
2
ν
m
,
1
-
α
˙
1
.
Define the functionβ
i (2
≤
i
≤
ρ) as
(42)
β
i
=
∂
α
i
-
1
∂
y
(
ω
0
+
ω
T
θ
^
)
+
∂
α
i
-
1
∂
ξ
0
ξ
˙
0
+
∂
α
i
-
1
∂
k
k
˙
+
∑
j
=
1
q
∂
α
i
-
1
∂
ξ
j
ξ
˙
j
+
∑
j
=
1
q
∑
i
=
1
m
+
1
∂
α
i
-
1
∂
ς
i
,
j
ς
˙
i
,
j
+
∑
j
=
1
i
∂
α
i
-
1
∂
y
r
(
j
-
1
)
y
r
(
j
)
+
∑
j
=
1
m
+
1
∂
α
i
-
1
∂
λ
j
λ
˙
j
.
Definez
3
=
ν
m
,
3
-
α
2 and α
2 is chosen as
(43)
α
2
=
-
κ
^
m
z
1
-
c
2
z
2
-
l
1
+
2
μ
(
∂
α
1
∂
y
)
2
z
2
-
β
2
+
∂
α
1
∂
ψ
^
m
r
ψ
τ
2
ψ
+
q
2
l
2
ν
m
,
1
+
∂
α
1
∂
θ
^
Γ
τ
2
-
ψ
^
m
∂
α
1
∂
y
tanh
(
z
2
(
∂
α
1
/
∂
y
)
δ
2
)
,
τ
2
=
τ
1
-
∂
α
1
∂
y
Γ
-
1
ω
z
2
+
Γ
-
1
[
0
1
×
p
,
z
1
z
2
,
0
,
…
,
0
]
T
,
τ
2
ψ
=
τ
1
ψ
+
z
2
∂
α
1
∂
y
tanh
(
z
2
(
∂
α
1
/
∂
y
)
δ
2
)
.
The time derivative ofz
2 along with (41)–(43) is
(44)
z
˙
2
=
z
3
-
κ
^
m
z
1
-
c
2
z
2
-
∂
α
1
∂
y
θ
~
T
ω
-
l
1
+
2
μ
(
∂
α
1
∂
y
)
2
z
2
-
∂
α
1
∂
y
x
~
2
+
∂
α
1
∂
θ
^
(
τ
2
-
θ
^
˙
)
+
∂
α
1
∂
ψ
^
m
(
τ
2
ψ
-
ψ
^
˙
m
)
-
ψ
^
m
∂
α
1
∂
y
tanh
(
z
2
(
∂
α
1
/
∂
y
)
δ
2
)
.
Consider the Lyapunov function candidate as(45)
V
2
=
V
1
+
1
2
z
2
2
.
The time derivative ofV
2 along with (42) is
(46)
V
˙
2
≤
z
2
z
3
-
c
1
z
1
2
-
c
2
z
2
2
+
1
2
l
x
-
2
2
+
θ
~
T
Γ
(
τ
1
-
θ
^
˙
)
+
κ
m
z
1
N
(
k
)
τ
+
k
˙
+
z
2
∂
α
1
∂
θ
^
(
τ
2
-
θ
^
˙
)
+
σ
ψ
ψ
~
m
T
ψ
^
m
+
1
r
ψ
ψ
~
m
(
r
ψ
τ
1
ψ
-
ψ
^
˙
m
)
+
σ
θ
θ
~
T
θ
^
+
z
2
∂
α
1
∂
ψ
^
m
(
τ
2
ψ
-
ψ
^
˙
m
)
+
∑
j
=
1
2
0.2785
δ
j
ψ
m
.Step i (3
≤
i
≤
ρ
-
1).
The time derivative ofz
i along with (31) is
(47)
z
˙
i
=
ν
m
,
i
-
q
i
l
i
ν
m
,
1
-
α
˙
i
-
1
.
Choose stabilizing functionα
i as
(48)
α
i
=
-
z
i
-
1
-
c
i
z
i
-
l
1
+
2
μ
(
∂
α
i
-
1
∂
y
)
2
z
i
+
q
i
l
i
ν
m
,
1
-
β
i
+
∂
α
i
-
1
∂
θ
^
Γ
τ
i
+
∂
α
i
-
1
∂
k
k
˙
+
∂
α
i
-
1
∂
ψ
^
m
r
ψ
τ
i
ψ
-
ψ
^
m
∂
α
i
-
1
∂
y
×
tanh
(
z
i
(
∂
α
i
-
1
/
∂
y
)
δ
i
)
-
(
∑
k
=
2
i
-
1
z
k
∂
α
k
-
1
∂
θ
^
)
Γ
∂
α
i
-
1
∂
y
ω
-
(
∑
k
=
2
i
-
1
z
k
∂
α
k
-
1
∂
ψ
^
m
)
γ
ψ
∂
α
i
-
1
∂
y
tanh
(
z
i
(
∂
α
i
-
1
/
∂
y
)
δ
i
)
,
τ
i
=
τ
i
-
1
-
∂
α
i
-
1
∂
y
Γ
-
1
ω
z
i
,
τ
i
ψ
=
τ
(
i
-
1
)
ψ
+
z
i
∂
α
i
-
1
∂
y
tanh
(
z
i
(
∂
α
i
-
1
/
∂
y
)
δ
i
)
.
Consider the Lyapunov function candidate as(49)
V
i
=
V
i
-
1
+
1
2
z
i
2
.
The time derivative ofV
i along with (47) and (48) is
(50)
V
˙
i
≤
z
i
z
i
+
1
-
∑
j
=
1
i
c
j
z
j
2
+
i
4
l
x
-
2
2
+
∑
j
=
2
i
z
j
∂
α
j
-
1
∂
θ
^
(
τ
j
-
θ
^
˙
)
+
θ
~
T
Γ
(
τ
i
-
θ
^
˙
)
+
∑
j
=
2
i
z
j
∂
α
j
-
1
∂
ψ
^
m
(
τ
j
ψ
-
ψ
^
˙
m
)
+
σ
ψ
ψ
~
m
T
ψ
^
m
+
1
r
ψ
ψ
~
m
(
r
ψ
τ
1
ψ
-
ψ
^
˙
m
)
-
∑
j
=
3
i
∑
k
=
2
j
-
1
z
j
∂
α
j
-
1
∂
θ
^
z
k
∂
α
k
-
1
∂
y
Γ
-
1
ω
-
∑
j
=
3
i
∑
k
=
2
j
-
1
z
j
∂
α
j
-
1
∂
ψ
^
m
z
k
∂
α
k
-
1
∂
y
tanh
(
z
j
(
∂
α
j
-
1
/
∂
y
)
δ
j
)
+
σ
θ
θ
~
T
θ
^
+
κ
m
z
1
N
(
k
)
τ
+
k
˙
+
∑
j
=
1
i
0.2785
δ
j
ψ
m
.
Definez
ρ
=
v
m
,
ρ
-
α
ρ
-
1 and the time derivative of z
ρ along with (31) is
(51)
z
˙
ρ
=
u
*
+
ν
m
,
ρ
+
1
-
q
ρ
l
ρ
ν
m
,
1
-
α
˙
ρ
-
1
.
Finally, the actual control signal designed as(52)
u
*
=
α
ρ
-
ν
m
,
ρ
+
1
.
Choose the tuning functions and parameter adaptation law as(53)
θ
^
˙
=
τ
ρ
,
ψ
^
˙
m
=
r
ψ
τ
ρ
ψ
.
To prepare for the stability analysis, a candidate Lyapunov function for the closed-loop system is chosen as(54)
V
ρ
=
V
ρ
-
1
+
1
2
z
ρ
2
+
V
x
.
The time derivative ofV
ρ along with (50) and (51) is
(55)
V
˙
ρ
≤
-
∑
i
=
1
ρ
c
j
z
j
2
-
1
2
l
∥
x
-
∥
2
+
κ
m
z
1
N
(
k
)
τ
+
k
˙
+
ρ
4
l
∥
x
-
2
∥
2
-
σ
10
2
θ
~
T
θ
~
-
σ
ψ
2
ψ
~
m
2
+
σ
ψ
2
ψ
m
2
+
∥
P
L
-
1
∥
2
ψ
m
2
+
13
2
σ
10
M
1
2
+
∑
i
=
1
ρ
0.2785
δ
i
ψ
m
≤
-
C
0
V
+
D
+
κ
m
z
1
N
(
k
)
τ
+
k
˙
,
where
(56)
C
0
=
min
⁡
{
c
1
,
…
,
c
ρ
,
σ
10
λ
max
⁡
(
Γ
)
,
σ
ψ
2
,
l
λ
max
⁡
(
P
)
}
,
D
=
[
∑
i
=
1
ρ
0.2785
δ
i
ψ
m
13
2
σ
10
M
1
2
+
σ
ψ
2
ψ
m
2
+
ρ
4
l
∥
x
-
2
∥
2
+
∑
i
=
1
ρ
0.2785
δ
i
ψ
m
+
∥
P
L
-
1
∥
2
ψ
m
2
]
.
Multiplying (55) by exp
⁡
(
C
0
t
) yields and integrating (55) over [
0
,
t
], we have
(57)
V
ρ
≤
V
ρ
(
0
)
e
-
C
0
t
+
e
-
C
0
t
∫
0
t
[
κ
m
N
(
k
)
+
1
]
k
˙
e
C
0
τ
d
τ
+
∫
0
t
D
e
-
C
0
(
t
-
τ
)
d
τ
.
Next, at timet, p
1 actuator failures occur, which results in an abrupt change of θ, owing to the change of values of these parameters is finite. Moreover, from (28) and (30), we have
(58)
∫
0
t
D
e
-
C
0
(
t
-
τ
)
d
τ
which is bounded on [
0
,
t
]. Let C
d and C
N be the upper bound of ∫
0
t
D
e
-
C
0
(
t
-
τ
)
d
τ and e
-
C
0
t
∫
0
t
[
κ
m
N
(
k
)
+
1
]
k
˙
e
C
0
τ
d
τ
(59)
C
d
=
sup
⁡
t
∈
[
0
,
t
]
(
∫
0
t
D
e
-
C
0
(
t
-
τ
)
d
τ
)
,
(60)
C
N
=
sup
⁡
t
∈
[
0
,
t
]
(
e
-
C
0
t
∫
0
t
[
κ
m
N
(
k
)
+
1
]
k
˙
e
C
0
τ
d
τ
)
.
From (58) and (59), we have
(61)
0
≤
V
ρ
(
t
)
≤
V
ρ
(
0
)
+
C
d
C
0
+
e
-
C
0
t
∫
0
t
[
κ
m
N
(
k
)
+
1
]
k
˙
e
C
0
τ
d
τ
.According to Lemma4, we have V
ρ
(
t
), k
(
t
) and ∫
0
t
κ
m
N
(
k
)
k
˙
d
τ bound on [
0
,
t
). Therefore, z
i
(
t
)
,
…
,
z
ρ
,
N
(
k
) are bound on [
0
,
t
) for all t
>
0, and all signals in the closed-loop system are bounded on [
0
,
t
) for all t
>
0. According to the discussion in [37], we see that the above conclusion is true for t
=
+
∞. Thus, we know that x
i
,
z
i are semiglobally uniformly ultimately bounded, we also have inequalities (61) as well as
(62)
|
z
1
|
≤
2
(
V
ρ
(
0
)
-
C
d
C
0
)
e
-
C
0
t
+
2
(
C
d
C
0
+
C
N
)
.Choosing appropriate positive matrixΓ such that λ
min
⁡
(
Γ
)
>
0. Furthermore, in order to achieve the tracking error convergent to a small neighborhood around zero, the parameters c
i, σ
ψ and Γ should be chosen appropriately to make C
N as small as desired. In this sense, we have guaranteed transient response. This result of transient response of the system is a direct consequence of the underlying robust filter structure of the ARFTC controller.


## 3.1. State Estimation

Control signalsu
j
(
t
) are designed such that u
*
=
u
j
β
(
y
). With fault model (2) and the chosen actuation scheme, we can rewrite the control inputs as follows:
(10)
u
j
(
t
)
=
η
j
β
j
(
y
)
(
1
-
σ
j
)
u
*
(
t
)
+
σ
j
u
-
j
,
j
=
1
,
…
,
q
,
where σ
j
=
1 corresponds to stuck actuator, and σ
j
=
0, η
j
∈
[
(
η
j
)
min
⁡
,
1
] represents efficiency loss of the actuator. In accordance with this, we rewrite the system as follows:
(11)
x
˙
=
A
x
+
φ
0
(
y
)
+
W
*
T
𝚽
(
y
)
+
F
(
y
)
x
+
∑
i
=
0
m
e
n
-
i
κ
i
u
*
+
∑
j
=
1
q
∑
i
=
0
m
e
n
-
i
μ
i
,
j
β
j
(
y
)
+
Δ
(
y
,
t
)
,
y
=
c
T
x
,
where e
1 denotes the first coordinate vector in R
n
(12)
A
=
[
0
⋮
I
n
-
1
0
0
]
,
Φ
(
y
)
=
[
Φ
1
(
y
)
0
⋯
0
0
Φ
2
(
y
)
⋱
⋮
⋮
⋮
⋱
0
0
0
⋯
Φ
n
(
y
)
]
,
F
(
y
)
=
[
0
0
⋯
0
0
f
2,2
⋱
⋮
⋮
⋮
⋱
0
0
f
2,2
⋯
f
n
,
n
]
,
φ
0
(
y
)
=
[
φ
0,1
(
y
)
φ
0,2
(
y
)
⋮
φ
0
,
n
(
y
)
]
,
Δ
=
[
d
1
+
ε
1
d
2
+
ε
2
⋮
d
n
+
ε
n
]
,
κ
i
=
∑
j
=
1
q
η
j
(
1
-
σ
j
)
b
i
,
j
,
μ
i
,
j
=
σ
j
u
-
j
b
i
,
j
,
c
=
[
1
0
⋯
0
]
T
,
W
*
=
[
W
1
*
W
2
*
⋯
W
n
*
]
,
where i
=
0
,
1
,
…
,
m, j
=
1
,
2
,
…
,
q. It can be deduced from Assumption 3 that |
Δ
(
x
,
t
)
|
≤
ε
i
+
d
i
<
ψ
m, where ψ
m is an unknown bounded constant.Note thatκ
i is the unknown measure of actuator effectiveness after faults and μ
i
,
j is the unknown measure of the fault magnitude which needs to be compensated.Thus, the states of system (1), unknown constants and parameter vectors, should be estimated by using the filters given in [26, 27]. We will define the following set of filters for the purpose of state-estimation:
(13)
ξ
˙
0
=
(
A
-
l
L
q
c
T
)
ξ
0
+
l
L
q
y
+
φ
0
(
y
)
+
F
(
y
)
ξ
0
,
ξ
0
∈
R
n
×
1
ξ
˙
=
(
A
-
l
L
q
c
T
)
ξ
+
Φ
(
y
)
,
ξ
∈
R
n
×
l
n
ν
˙
i
=
(
A
-
l
L
q
c
T
)
ν
i
+
e
n
-
i
u
*
+
F
(
y
)
ν
i
,
ν
i
∈
R
n
×
1
ψ
˙
i
,
j
=
(
A
-
l
L
q
c
T
)
ψ
i
,
j
+
e
n
-
i
β
j
(
y
)
,
ψ
i
,
j
∈
R
n
×
1
,
where the gain matrix q
=
[
q
1
,
q
2
,
…
,
q
n
]
T is chosen to make A
-
q
c
T Hurwitz, i
=
0
,
1
,
…
,
m, j
=
1
,
2
,
…
,
q, L
=
diag
⁡
[
1
l
⋯
l
n
-
1
], and l is the observer gain updated by
(14)
l
˙
=
-
κ
l
2
+
κ
l
+
l
γ
(
y
)
,
l
(
0
)
=
1
with κ a positive design parameter and γ
(
y
) a nonnegative smooth function. It can be proved by contradiction that l
(
t
)
≥
1 for all t
≥
0.Due to the special structure ofA, the order of K-filters can be reduced by using the following two filters:
(15)
λ
˙
=
(
A
-
l
L
q
c
T
)
λ
+
e
n
u
*
+
F
(
y
)
λ
,
ς
˙
j
=
(
A
-
l
L
q
c
T
)
ς
j
+
e
n
β
j
(
y
)
,
j
=
1,2
,
…
,
q
and the following algebraic equations:
(16)
ν
i
=
(
A
-
l
L
q
c
T
)
i
λ
,
ψ
i
,
j
=
(
A
-
l
L
q
c
T
)
i
ς
j
,
i
=
0,1
,
…
,
m
.The estimated state can be written as(17)
x
^
=
ξ
0
+
ξ
W
*
+
∑
i
=
0
m
κ
i
ν
i
+
∑
j
=
1
q
∑
i
=
0
m
μ
i
,
j
ψ
i
,
j
.Letx
~
=
x
-
x
^ be the estimation error. Then, the state estimation error dynamic is given by
(18)
x
~
˙
(
t
)
=
(
A
-
l
L
q
c
T
)
x
~
+
F
(
y
)
x
~
+
Δ
.Noting that the change of coordinatesx
-
=
l
-
μ
L
-
1
x
~ with  μ a positive design parameter (18) is transformed into
(19)
x
-
˙
=
l
(
A
-
q
c
T
)
x
-
+
L
-
1
F
(
y
)
L
x
-
+
l
-
μ
L
-
1
Δ
-
l
˙
l
(
μ
I
+
D
)
x
-
,
where D
=
diag
⁡
{
0
,
1
,
…
,
n
-
1
}. Since A
-
q
c
T is Hurwitz, there is a symmetric positive definite matrix P satisfying P
(
A
-
q
c
T
)
+
(
A
-
q
c
T
)
T
P
=
-
I. Let the quadratic Lyapunov function V
x
=
x
-
T
P
x
-, whose derivative is computed as
(20)
V
˙
x
=
-
l
∥
x
-
∥
2
+
2
x
-
T
P
L
-
1
F
(
y
)
L
x
-
+
2
l
-
μ
x
-
T
P
L
-
1
Δ
-
l
˙
l
x
-
T
(
P
D
+
D
P
+
2
μ
P
)
x
-
.Note thatl
≥
1. Then there is a nonnegative smooth function γ
1
(
y
) such that L
-
1
F
(
y
)
L
≤
γ
1
(
y
), from which it follows that
(21)
2
x
-
T
P
L
-
1
F
(
y
)
L
x
-
≤
2
∥
P
∥
γ
1
(
y
)
∥
x
-
∥
2
.From Young’s inequality, we have(22)
2
l
-
μ
x
-
T
P
L
-
1
Δ
≤
l
-
2
μ
x
-
T
x
-
+
∥
P
L
-
1
∥
2
ψ
m
2
.SinceP is a symmetric positive definite matrix, by choosing a sufficiently large μ, we can obtain
(23)
σ
1
I
≤
D
P
+
P
D
+
2
μ
P
≤
σ
2
I
which together with (14) implies that
(24)
-
l
˙
l
x
-
T
(
P
D
+
D
P
+
2
μ
P
)
x
-
≤
κ
σ
2
l
∥
x
-
∥
2
-
σ
1
γ
(
y
)
∥
x
-
∥
2
,
where σ
1, σ
2 are positive constants. From (21) and (24), (18) can be rewritten as
(25)
V
˙
x
≤
-
[
(
1
-
l
-
2
μ
-
κ
σ
2
)
l
+
σ
1
γ
(
y
)
aaaaaa
-
2
∥
P
∥
γ
1
(
y
)
l
-
2
μ
]
∥
x
-
∥
2
+
∥
P
L
-
1
∥
2
ψ
m
2
.
By choosing κ and γ
(
y
) to satisfy κ
≤
(
1
-
2
l
-
2
μ
)
/
(
2
σ
2
), γ
(
y
)
≥
2
∥
P
∥
γ
1
(
y
)
/
σ
1
≥
0, we arrive at
(26)
V
˙
x
≤
-
1
2
l
∥
x
-
∥
2
+
∥
P
L
-
1
∥
2
ψ
m
2
.


## 3.2. Parameter Estimate

Letθ
^ denote the estimate of θ and θ
~
=
θ
-
θ
^ denote the estimation error. The extent of parametric uncertainties satisfy
(27)
Ω
=
{
θ
∣
∥
θ
∥
≤
M
1
}
,
where M
1 is a positive design parameter.It is well known that parameter estimation algorithms suffer from parameter drift in presence of disturbances, resulting in system states growing unboundedly. We use the switching functionσ algorithm [36] to deal with this problem. The update law used here has the following form:
(28)
σ
θ
(
∥
θ
^
∥
)
=
{
0
,
∥
θ
^
∥
≤
M
1
χ
1
(
∥
θ
^
∥
)
,
M
1
≤
∥
θ
^
∥
≤
2
M
1
σ
10
,
∥
θ
^
∥
≥
2
M
1
,
where σ
10 is a positive design parameter, and χ
1 is an arbitrary adaptation function.Consider(29)
χ
1
(
∥
θ
^
∥
)
=
σ
10
(
2
ρ
-
1
)
!
M
1
2
ρ
-
1
[
(
ρ
-
1
)
!
]
2
∫
M
1
∥
θ
^
∥
(
Θ
-
M
1
)
(
2
M
1
-
Θ
)
d
Θ
.The mapping guarantees that the following properties are always satisfied:(30)
σ
θ
θ
~
T
θ
^
≤
-
1
2
σ
10
θ
~
T
θ
~
+
13
2
σ
10
M
1
2
.


## 3.3. Controller Design

Furthermore, system (13) can be represented as
(31)
ν
˙
m
,
i
=
ν
˙
m
,
i
+
1
-
q
i
l
i
ν
˙
m
,
1
,
i
=
2,3
,
…
,
ρ
-
1
,
ν
˙
m
,
ρ
=
ν
˙
m
,
ρ
+
1
-
q
ρ
l
ρ
ν
˙
m
,
1
+
u
*
.The derivative of the outputy is given by
(32)
y
˙
=
ω
0
+
ω
T
θ
+
Δ
1
(
y
,
t
)
=
κ
m
ν
m
,
2
+
ω
0
+
ω
-
T
θ
+
Δ
1
(
y
,
t
)
,
where the regressor ω and truncated regressor ω
0 are defined as [26]
(33)
ω
0
=
[
ξ
0,2
+
φ
0,1
]
,
ω
=
[
ψ
0
,
q
(
2
)
,
ξ
(
2
)
+
𝚽
(
1
)
,
ν
m
,
2
,
ν
m
-
1,2
,
…
,
ν
0,2
,
a
a
i
a
ψ
m
,
1
(
2
)
,
…
,
ψ
m
,
q
(
2
)
,
…
,
ψ
0,1
(
2
)
,
…
,
ψ
0
,
q
(
2
)
,
]
T
,
ω
-
=
ω
-
e
l
n
+
1
ν
m
,
2
θ
=
[
W
*
,
κ
m
,
…
,
κ
0
,
μ
m
,
1
,
…
,
μ
m
,
q
,
…
,
μ
0,1
,
…
,
μ
0
,
q
]
T
.In this section, we present the adaptive output-feedback control design using the backstepping technique. Define the following error coordinates:z
1
=
y
-
y
d and z
i
=
v
m
,
i
-
α
i
-
1, i
=
2,3
,
…
,
ρ, where α
i
-
1 is the stabilizing functions to be designed.Step 1.
Differentiatingz
1 with respect to time t, we obtain
(34)
z
˙
1
=
κ
m
ν
m
,
2
+
ω
0
+
ω
-
T
θ
-
y
˙
d
+
Δ
1
(
y
,
t
)
.
The problem of the unknown sign of the virtual direction is sloved by the Nussbaum-type functionsκ
m. Choose the tuning functions and parameter adaptation law as
(35)
α
1
=
N
(
k
)
τ
,
k
˙
=
z
1
τ
,
τ
=
c
1
z
1
+
ω
-
T
θ
^
-
y
˙
d
+
l
1
+
2
μ
z
1
+
ψ
^
m
tanh
⁡
(
z
1
δ
1
)
,
where N
(
k
) is Nussbaum gain; ψ
^
m is an estimate of ψ
m with the estimation error ψ
~
m
=
ψ
m
-
ψ
^
m; c
1 is a positive constant. Using the inequality
(36)
0
≤
|
z
1
|
-
z
1
tanh
(
z
1
δ
)
≤
0.2785
δ
,
where δ is a positive design parameter and substituting (35) and (36) into (34) yield
(37)
z
˙
1
=
κ
m
[
z
2
+
N
(
k
)
τ
]
+
Δ
1
(
y
,
t
)
+
θ
~
T
Γ
(
τ
1
-
θ
^
˙
)
+
σ
θ
θ
~
T
θ
^
+
τ
-
c
1
z
1
-
l
1
+
2
μ
z
1
-
ψ
^
m
tanh
(
z
1
δ
1
)
,
where τ
1
=
z
1
Γ
-
1
ω
-
-
σ
θ
(
∥
θ
^
∥
)
Γ
-
1
θ
^.
Define the quadratic function(38)
V
1
=
1
2
z
1
2
+
1
2
θ
~
T
Γ
θ
~
+
1
2
r
ψ
ψ
~
m
2
.
From Young’s inequality, we obtain(39)
z
1
x
~
2
≤
l
1
+
2
μ
z
1
2
+
l
x
-
2
2
4
.
DifferentiatingV
1 with respect to time t leads to
(40)
V
˙
1
=
κ
m
z
1
z
2
-
c
1
z
1
2
+
κ
m
z
1
N
(
k
)
τ
+
k
˙
+
1
4
l
x
-
2
2
+
θ
~
T
Γ
(
τ
1
-
θ
^
˙
)
+
σ
θ
θ
~
T
θ
^
+
σ
ψ
ψ
~
m
T
ψ
^
m
+
1
r
ψ
ψ
~
m
(
r
ψ
τ
1
ψ
-
ψ
^
˙
m
)
+
0.2785
δ
1
ψ
m
,
where τ
1
ψ
=
z
1
tanh
(
z
1
/
δ
1
)
-
σ
ψ
ψ
^
m.Step 2.
The time derivative ofz
2 along with (31) is
(41)
z
˙
2
=
ν
m
,
3
-
q
2
l
2
ν
m
,
1
-
α
˙
1
.
Define the functionβ
i (2
≤
i
≤
ρ) as
(42)
β
i
=
∂
α
i
-
1
∂
y
(
ω
0
+
ω
T
θ
^
)
+
∂
α
i
-
1
∂
ξ
0
ξ
˙
0
+
∂
α
i
-
1
∂
k
k
˙
+
∑
j
=
1
q
∂
α
i
-
1
∂
ξ
j
ξ
˙
j
+
∑
j
=
1
q
∑
i
=
1
m
+
1
∂
α
i
-
1
∂
ς
i
,
j
ς
˙
i
,
j
+
∑
j
=
1
i
∂
α
i
-
1
∂
y
r
(
j
-
1
)
y
r
(
j
)
+
∑
j
=
1
m
+
1
∂
α
i
-
1
∂
λ
j
λ
˙
j
.
Definez
3
=
ν
m
,
3
-
α
2 and α
2 is chosen as
(43)
α
2
=
-
κ
^
m
z
1
-
c
2
z
2
-
l
1
+
2
μ
(
∂
α
1
∂
y
)
2
z
2
-
β
2
+
∂
α
1
∂
ψ
^
m
r
ψ
τ
2
ψ
+
q
2
l
2
ν
m
,
1
+
∂
α
1
∂
θ
^
Γ
τ
2
-
ψ
^
m
∂
α
1
∂
y
tanh
(
z
2
(
∂
α
1
/
∂
y
)
δ
2
)
,
τ
2
=
τ
1
-
∂
α
1
∂
y
Γ
-
1
ω
z
2
+
Γ
-
1
[
0
1
×
p
,
z
1
z
2
,
0
,
…
,
0
]
T
,
τ
2
ψ
=
τ
1
ψ
+
z
2
∂
α
1
∂
y
tanh
(
z
2
(
∂
α
1
/
∂
y
)
δ
2
)
.
The time derivative ofz
2 along with (41)–(43) is
(44)
z
˙
2
=
z
3
-
κ
^
m
z
1
-
c
2
z
2
-
∂
α
1
∂
y
θ
~
T
ω
-
l
1
+
2
μ
(
∂
α
1
∂
y
)
2
z
2
-
∂
α
1
∂
y
x
~
2
+
∂
α
1
∂
θ
^
(
τ
2
-
θ
^
˙
)
+
∂
α
1
∂
ψ
^
m
(
τ
2
ψ
-
ψ
^
˙
m
)
-
ψ
^
m
∂
α
1
∂
y
tanh
(
z
2
(
∂
α
1
/
∂
y
)
δ
2
)
.
Consider the Lyapunov function candidate as(45)
V
2
=
V
1
+
1
2
z
2
2
.
The time derivative ofV
2 along with (42) is
(46)
V
˙
2
≤
z
2
z
3
-
c
1
z
1
2
-
c
2
z
2
2
+
1
2
l
x
-
2
2
+
θ
~
T
Γ
(
τ
1
-
θ
^
˙
)
+
κ
m
z
1
N
(
k
)
τ
+
k
˙
+
z
2
∂
α
1
∂
θ
^
(
τ
2
-
θ
^
˙
)
+
σ
ψ
ψ
~
m
T
ψ
^
m
+
1
r
ψ
ψ
~
m
(
r
ψ
τ
1
ψ
-
ψ
^
˙
m
)
+
σ
θ
θ
~
T
θ
^
+
z
2
∂
α
1
∂
ψ
^
m
(
τ
2
ψ
-
ψ
^
˙
m
)
+
∑
j
=
1
2
0.2785
δ
j
ψ
m
.Step i (3
≤
i
≤
ρ
-
1).
The time derivative ofz
i along with (31) is
(47)
z
˙
i
=
ν
m
,
i
-
q
i
l
i
ν
m
,
1
-
α
˙
i
-
1
.
Choose stabilizing functionα
i as
(48)
α
i
=
-
z
i
-
1
-
c
i
z
i
-
l
1
+
2
μ
(
∂
α
i
-
1
∂
y
)
2
z
i
+
q
i
l
i
ν
m
,
1
-
β
i
+
∂
α
i
-
1
∂
θ
^
Γ
τ
i
+
∂
α
i
-
1
∂
k
k
˙
+
∂
α
i
-
1
∂
ψ
^
m
r
ψ
τ
i
ψ
-
ψ
^
m
∂
α
i
-
1
∂
y
×
tanh
(
z
i
(
∂
α
i
-
1
/
∂
y
)
δ
i
)
-
(
∑
k
=
2
i
-
1
z
k
∂
α
k
-
1
∂
θ
^
)
Γ
∂
α
i
-
1
∂
y
ω
-
(
∑
k
=
2
i
-
1
z
k
∂
α
k
-
1
∂
ψ
^
m
)
γ
ψ
∂
α
i
-
1
∂
y
tanh
(
z
i
(
∂
α
i
-
1
/
∂
y
)
δ
i
)
,
τ
i
=
τ
i
-
1
-
∂
α
i
-
1
∂
y
Γ
-
1
ω
z
i
,
τ
i
ψ
=
τ
(
i
-
1
)
ψ
+
z
i
∂
α
i
-
1
∂
y
tanh
(
z
i
(
∂
α
i
-
1
/
∂
y
)
δ
i
)
.
Consider the Lyapunov function candidate as(49)
V
i
=
V
i
-
1
+
1
2
z
i
2
.
The time derivative ofV
i along with (47) and (48) is
(50)
V
˙
i
≤
z
i
z
i
+
1
-
∑
j
=
1
i
c
j
z
j
2
+
i
4
l
x
-
2
2
+
∑
j
=
2
i
z
j
∂
α
j
-
1
∂
θ
^
(
τ
j
-
θ
^
˙
)
+
θ
~
T
Γ
(
τ
i
-
θ
^
˙
)
+
∑
j
=
2
i
z
j
∂
α
j
-
1
∂
ψ
^
m
(
τ
j
ψ
-
ψ
^
˙
m
)
+
σ
ψ
ψ
~
m
T
ψ
^
m
+
1
r
ψ
ψ
~
m
(
r
ψ
τ
1
ψ
-
ψ
^
˙
m
)
-
∑
j
=
3
i
∑
k
=
2
j
-
1
z
j
∂
α
j
-
1
∂
θ
^
z
k
∂
α
k
-
1
∂
y
Γ
-
1
ω
-
∑
j
=
3
i
∑
k
=
2
j
-
1
z
j
∂
α
j
-
1
∂
ψ
^
m
z
k
∂
α
k
-
1
∂
y
tanh
(
z
j
(
∂
α
j
-
1
/
∂
y
)
δ
j
)
+
σ
θ
θ
~
T
θ
^
+
κ
m
z
1
N
(
k
)
τ
+
k
˙
+
∑
j
=
1
i
0.2785
δ
j
ψ
m
.
Definez
ρ
=
v
m
,
ρ
-
α
ρ
-
1 and the time derivative of z
ρ along with (31) is
(51)
z
˙
ρ
=
u
*
+
ν
m
,
ρ
+
1
-
q
ρ
l
ρ
ν
m
,
1
-
α
˙
ρ
-
1
.
Finally, the actual control signal designed as(52)
u
*
=
α
ρ
-
ν
m
,
ρ
+
1
.
Choose the tuning functions and parameter adaptation law as(53)
θ
^
˙
=
τ
ρ
,
ψ
^
˙
m
=
r
ψ
τ
ρ
ψ
.
To prepare for the stability analysis, a candidate Lyapunov function for the closed-loop system is chosen as(54)
V
ρ
=
V
ρ
-
1
+
1
2
z
ρ
2
+
V
x
.
The time derivative ofV
ρ along with (50) and (51) is
(55)
V
˙
ρ
≤
-
∑
i
=
1
ρ
c
j
z
j
2
-
1
2
l
∥
x
-
∥
2
+
κ
m
z
1
N
(
k
)
τ
+
k
˙
+
ρ
4
l
∥
x
-
2
∥
2
-
σ
10
2
θ
~
T
θ
~
-
σ
ψ
2
ψ
~
m
2
+
σ
ψ
2
ψ
m
2
+
∥
P
L
-
1
∥
2
ψ
m
2
+
13
2
σ
10
M
1
2
+
∑
i
=
1
ρ
0.2785
δ
i
ψ
m
≤
-
C
0
V
+
D
+
κ
m
z
1
N
(
k
)
τ
+
k
˙
,
where
(56)
C
0
=
min
⁡
{
c
1
,
…
,
c
ρ
,
σ
10
λ
max
⁡
(
Γ
)
,
σ
ψ
2
,
l
λ
max
⁡
(
P
)
}
,
D
=
[
∑
i
=
1
ρ
0.2785
δ
i
ψ
m
13
2
σ
10
M
1
2
+
σ
ψ
2
ψ
m
2
+
ρ
4
l
∥
x
-
2
∥
2
+
∑
i
=
1
ρ
0.2785
δ
i
ψ
m
+
∥
P
L
-
1
∥
2
ψ
m
2
]
.
Multiplying (55) by exp
⁡
(
C
0
t
) yields and integrating (55) over [
0
,
t
], we have
(57)
V
ρ
≤
V
ρ
(
0
)
e
-
C
0
t
+
e
-
C
0
t
∫
0
t
[
κ
m
N
(
k
)
+
1
]
k
˙
e
C
0
τ
d
τ
+
∫
0
t
D
e
-
C
0
(
t
-
τ
)
d
τ
.
Next, at timet, p
1 actuator failures occur, which results in an abrupt change of θ, owing to the change of values of these parameters is finite. Moreover, from (28) and (30), we have
(58)
∫
0
t
D
e
-
C
0
(
t
-
τ
)
d
τ
which is bounded on [
0
,
t
]. Let C
d and C
N be the upper bound of ∫
0
t
D
e
-
C
0
(
t
-
τ
)
d
τ and e
-
C
0
t
∫
0
t
[
κ
m
N
(
k
)
+
1
]
k
˙
e
C
0
τ
d
τ
(59)
C
d
=
sup
⁡
t
∈
[
0
,
t
]
(
∫
0
t
D
e
-
C
0
(
t
-
τ
)
d
τ
)
,
(60)
C
N
=
sup
⁡
t
∈
[
0
,
t
]
(
e
-
C
0
t
∫
0
t
[
κ
m
N
(
k
)
+
1
]
k
˙
e
C
0
τ
d
τ
)
.
From (58) and (59), we have
(61)
0
≤
V
ρ
(
t
)
≤
V
ρ
(
0
)
+
C
d
C
0
+
e
-
C
0
t
∫
0
t
[
κ
m
N
(
k
)
+
1
]
k
˙
e
C
0
τ
d
τ
.According to Lemma4, we have V
ρ
(
t
), k
(
t
) and ∫
0
t
κ
m
N
(
k
)
k
˙
d
τ bound on [
0
,
t
). Therefore, z
i
(
t
)
,
…
,
z
ρ
,
N
(
k
) are bound on [
0
,
t
) for all t
>
0, and all signals in the closed-loop system are bounded on [
0
,
t
) for all t
>
0. According to the discussion in [37], we see that the above conclusion is true for t
=
+
∞. Thus, we know that x
i
,
z
i are semiglobally uniformly ultimately bounded, we also have inequalities (61) as well as
(62)
|
z
1
|
≤
2
(
V
ρ
(
0
)
-
C
d
C
0
)
e
-
C
0
t
+
2
(
C
d
C
0
+
C
N
)
.Choosing appropriate positive matrixΓ such that λ
min
⁡
(
Γ
)
>
0. Furthermore, in order to achieve the tracking error convergent to a small neighborhood around zero, the parameters c
i, σ
ψ and Γ should be chosen appropriately to make C
N as small as desired. In this sense, we have guaranteed transient response. This result of transient response of the system is a direct consequence of the underlying robust filter structure of the ARFTC controller.


## 4. Application Example

To demonstrate the effectiveness of the proposed approach, we consider the following nonlinear system:(63)
x
˙
1
=
x
2
+
f
1
(
y
)
+
a
1
y
+
a
2
sin
(
y
)
+
Δ
1
(
x
,
t
)
,
x
˙
2
=
a
3
y
2
+
a
4
y
+
y
x
2
+
f
2
(
y
)
+
b
01
β
1
(
y
)
u
1
+
b
02
β
2
(
y
)
u
2
+
Δ
2
(
x
,
t
)
,
y
=
x
1
,
where a
1
=
12, a
2
=
-
1, a
3
=
1, a
4
=
-
1, b
01
=
1, b
02
=
0.3, β
1
(
y
)
=
β
2
(
y
)
=
1
+
|
y
cos
⁡
(
y
)
|, f
1
(
y
)
=
3
x
2
y
2, and f
2
(
y
)
=
y
2. In the simulation, RBFNNs are applied and we select the centers and widths as neural networks W
^
1
T
𝚽
1
(
y
) contain 25 nodes (i.e., l
1
=
25) with centers c
l
(
l
=
1,2
,
…
,
l
1
) evenly spaced in [
-
4.0,4.0
]
×
[
-
4.0,4.0
] and width centers η
l
=
1.0 (l
=
1,2
,
…
,
l
1), and 25 nodes (i.e., l
2
=
25) with centers c
l
(
l
=
1,2
,
…
,
l
2
) spaced in [
-
4.0,4.0
]
×
[
-
4.0,4.0
]
×
[
-
4.0,4.0
]
×
[
-
4.0,4.0
] and widths η
l
=
1.0
(
l
=
1,2
,
…
,
l
2
) for NN W
^
2
T
𝚽
2
(
y
).Note that the same reduced order model was used by Tang et al. [19] and thus will provide a platform to compare the robust adaptive control (RAC) based fault-tolerant control (FTC) with the adaptive robust fault-tolerant control (ARFTC) schemes.Case 1 (no failures occur).
The simulation results are shown in Figures1 and 2 for y
d
=
sin
(
0.5
t
). In the first set of simulations (see Figure 1), all unstructured modeling errors and disturbances are assumed to be zero; that is, Δ
i
=
0 for i
=
1
,
2. The initial conditions x
=
[
0
1
]
T. The simulation parameters are as follows:
(64)
Γ
=
diag
⁡
{
0.5
0.5
10
10
5
5
1
1
1
1
1
}
,
q
=
[
1
1
]
T
,
r
ψ
=
1.5
,
μ
=
1
,
κ
=
0.5
,
δ
1
=
1
,
δ
2
=
0.5
,
c
1
=
12
,
c
2
=
5
,
k
(
0
)
=
0
,
γ
(
y
)
=
1.6
y
2
,
θ
^
(
0
)
=
diag
⁡
{
0
0
0
0
0
0
5
-

0.4
0.4
-

0.5
0.5
}
,
N
(
k
)
=
k
2
cos
⁡
(
k
)
.Figure 1
Plant output and reference signal.Figure 2
Nussbaum gainN
(
k
) and its argument k.Case 2.
Actuator faults occur and the actual control signal is given by(65)
u
1
(
t
)
=
{
ν
1
(
t
)
,
t
∈
(
0,30
)
u
-
1
+
d
-
11
f
11
(
t
)
,
t
∈
(
30
,
∞
)
,
u
2
(
t
)
=
ν
2
(
t
)
,
t
∈
(
0
,
∞
)
.
Failure parameters are chosen to be u
-
1
=
6, d
11
=
2, f
11
(
t
)
=
0.5
sin
(
t
).The reference command is chosen asy
d
(
t
)
=
sin
(
0
.
5
t
). Details of RAC based FTC can be obtained from [19]. Additionally, the controller parameters were chosen such that the control input profiles would be comparable for both schemes. Two cases are considered in order to illustrate the effectiveness of the proposed scheme.In the first set of simulations,Δ
i
=
0 for i
=
1,2. Then, from Figures 1 and 2, we conclude that the presented control scheme can work effectively for nonlinear systems without failures in actuators. Furthermore, from Figure 3, both the systems perform well initially and have similar control input profiles. With the actuator failure, which causes a bigger jump in the parameter value, the tracking error stays close to zero in ARFTC based scheme but deviates significantly in the RAC based scheme. This can be explained as follows. The design of the robust component of the ARFTC control law has already incorporated such jumps in parameter values, and hence, it is better suited to handle the parametric uncertainties introduced due to actuator failures.Tracking error and control signals for ARFTC versus RAC based fault-tolerant schemes.
(a)
(b)
(c)The second set of simulations (see Figure4) is performed where disturbances were introduced to test the performance of the two schemes in presence of unknown bounded uncertainties. We set Δ
1
=
0.5
sin
(
3
t
) and Δ
2
=
0.3
sin
(
5
t
). The parametric uncertainty and unstructured uncertainty bounds are incorporated in the design of the baseline robust controller in ARFTC, resulting in guaranteeing desired transient response and acceptable steady-state tracking error. The performance of the RAC based scheme deteriorates significantly in presence of unstructured modeling uncertainties, which are inherent in any realistic system model. Therefore, the achievable performance using the proposed scheme is superior to that of RAC based schemes.Tracking error and control signals for ARFTC versus RAC based fault-tolerant schemes in presence of disturbances.
(a)
(b)
(c)


## 5. Conclusions

In this paper, an adaptive actuator failure compensation scheme based on the robust fault-tolerant control approach has been proposed. A linearly parameterized model with unknown parameters and actuator failure parameters has been established. The Nussbaum gain approach has been exploited to relax the assumption on the control gain signs. The control scheme introduced switchingσ algorithms to ensure that the estimation of time-varying parameters is bounded. The designed controller does not require precise information of failures, thereby improving the engineering application value. Finally, simulation studies have shown that the proposed adaptive robust control law is effective.



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*Source: 217501-2014-04-08.xml*