# Adaptive Output Tracking Control for Nonlinear Systems with Failed Actuators and Aircraft Flight System Applications

Authors: Chuanjing Hou; Lisheng Hu; Yingwei Zhang
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

An adaptive failure compensation scheme using output feedback is proposed for a class of nonlinear systems with nonlinearities depending on the unmeasured states of systems. Adaptive high-gain K-filters are presented to suppress the nonlinearities while the proposed backstepping adaptive high-gain controller guarantees the stability of the closed-loop system and small tracking errors. Simulation results verify that the adaptive failure compensation scheme is effective.


---

## Body

## 1. Introduction

Actuator failure compensation has significant impact on control systems, such as aircraft flight systems and nuclear power systems. Due to the unknown failure patterns, times, and values, it has been an important and challenging research problem. Remarkable progress has been made in the area, such as adaptive control [1–10], sliding-mode based designs [11–14], switching based designs [15–17], fault-detection diagnosis based designs, fuzzy systems based designs, and neural-network based designs. Since adaptive designs use an adaptive controller to accommodate the uncertainties, it has been extensively employed. An adaptive state feedback controller is proposed to guarantee the performance requirements in presence of actuator failures [1–4]. Adaptive actuator failure compensation using output feedback is studied for a class of nonlinear systems [5–9, 14, 16, 18–21]. Unfortunately, if the systems have nonlinearities that cannot be bounded by any function of output, the existing methods fail to compensate the control systems. Therefore, new techniques need to be developed.In this paper, an adaptive output feedback controller for a class of nonlinear systems with actuator failures is discussed. Motivated by the fact that the nonlinearities satisfying the growth condition [15, 22–25] can be suppressed by a dynamic high-gain output feedback controller, a modified compensation scheme is proposed where adaptive high-gain K-filters and an adaptive high-gain controller are introduced. The main contribution of our paper is as follows. (
1
) We relax the condition imposed by Tang et al. in [5–7]. (
2
) An adaptive output feedback controller is developed with switching laws [15, 16]. (
3
) The parameters of the K-filters [26] and the controller are adaptively tuned online depending on system nonlinearities and actuator failures. By applying the backstepping technique, the robust controller is recursively constructed step by step. Parameter update laws are addressed to ensure closed-loop signal boundedness and small tracking errors. The simulation results are presented to demonstrate the effectiveness of the scheme proposed in this paper.The rest of the paper is outlined as follows. In Section2, the control problem is formulated. In Section 3, a robust adaptive compensation scheme with switching laws via the backstepping design is proposed. In Section 4, the stability analysis is presented. Two detailed simulation examples show the proposed scheme is effective in Section 5. Finally, this paper is concluded by Section 6.


## 2. Problem Formulation

Consider a class of nonlinear dynamic systems in the form of(1)
x
˙
i
=
x
i
+
1
+
φ
0
,
i
y
+
ϕ
i
x
+
a
T
φ
i
i
=
1,2
,
…
,
ρ
-
1
,
x
˙
ρ
=
x
ρ
+
1
+
φ
0
,
ρ
y
+
ϕ
ρ
x
+
a
T
φ
ρ
+
∑
j
=
1
m
b
n
∗
,
j
β
j
y
u
j
,
⋮
x
˙
n
-
1
=
x
n
+
φ
0
,
n
-
1
y
+
ϕ
n
-
1
x
+
a
T
φ
n
-
1
+
∑
j
=
1
m
b
1
,
j
β
j
y
u
j
,
x
˙
n
=
φ
0
,
n
y
+
ϕ
n
x
+
a
T
φ
n
+
∑
j
=
1
m
b
0
,
j
β
j
y
u
j
,
y
=
x
1
,where ρ is the relative degree of the system and u
j
∈
R, j
=
1,2
,
…
,
m, are the control inputs whose actuators may fail during system operation; x
-
i
=
x
1
,
x
2
,
…
,
x
i
T
i
=
1
,
…
,
n
-
1 are the state vector; a
=
[
a
1
,
…
,
a
q
]
T, b
r
,
j
(
r
=
0,1
,
…
,
n
∗
=
n
-
ρ
,
j
=
1,2
,
…
,
m
) are unknown constant parameters. Only the output y is available for measurement. φ
0
,
i
y
i
=
1,2
,
…
,
n, β
j
y
≠
0
(
f
o
r
∀
y
∈
R
,
j
=
1,2
,
…
,
m
), and φ
j
,
i
y
j
=
1,2
,
…
,
q
,
i
=
1,2
,
…
,
n are known smooth nonlinear functions; ϕ
i
x
(
i
=
1,2
,
…
,
n
) are continuous unknown functions satisfying the following assumption.Assumption 1.
Fori
=
1,2
,
…
,
n, there is an unknown constant c
≥
0 such that (2)
ϕ
i
x
≤
c
x
1
+
⋯
+
x
i
.We denote u
i as the input of the i
t
h
i
=
1
,
…
,
m actuator. Suppose the actuator failure can be modeled as(3)
u
i
=
ρ
i
v
i
+
u
k
i
,
∀
t
≥
t
i
F
,
ρ
i
u
k
i
=
0
,
i
=
1
,
…
,
m
,where ρ
i
∈
0,1, u
k
i and t
i
F are all unknown constants, and v
i
t
i
=
1,2
,
…
,
m are applied control inputs to be designed in Section 3. For different values of ρ
i, three types of failures are included as follows: (1)
ρ
i
=
1; the actuator works normally, namely, u
i
=
u
c
i, which is regarded as a failure-free actuator;
(2)
0
<
ρ
i
<
1; it implies u
i
=
ρ
i
u
c
i. The ith actuator is called partial loss of effectiveness (PLOE);
(3)
ρ
i
=
0; it indicates u
i
=
u
k
i. The ith actuator is called total loss of effectiveness (TLOE).Remark 2.
The values ofρ
i can change only from ρ
i
=
1 to some values with 0
≤
ρ
i
<
1. This means that possible changes from normal to any one of the failure cases are unidirectional. The uniqueness of t
i
F indicates that a failure occurs only once on the ith actuator.
Our objective is to design an output feedback controller for the nonlinear systems (1) with p unknown actuator failures when p changes at time instants t
k, k
=
1,2
,
…
,
q, such that the plant output y
t tracks a given reference signal y
r
t with up to ρ
t
h-order derivatives bounded as close as possible and that all closed-loop signals are bounded despite the presence of unknown actuator failures and unknown plant parameters.


## 3. Adaptive Compensation Control Scheme

The zero dynamics of system (1) with actuator failures are only dependent on the failure pattern [1]. For a fixed failure pattern, there is a resulting pattern of zero dynamics. Since the failure pattern j
=
j
1
,
…
,
j
p, 0
≤
p
≤
m
-
1 is unknown, a desirable adaptive design is expected to achieve the control objective for any possible failure pattern. For the closed-loop stability, all zero dynamics corresponding to the possible failure patterns need to be stable. To derive a suitable adaptive control scheme, the following assumptions are made.Assumption 3.
When TOLE type of actuator failures is up tom
-
1, the remaining actuators can still achieve a desired control objective.Assumption 4.
The polynomials∑
j
≠
j
1
,
j
2
,
…
,
j
p
m
s
i
g
n
[
b
n
∗
,
j
]
B
j
(
s
) are stable, (4)
∀
j
1
,
j
2
,
…
,
j
p
⊂
1,2
,
…
,
m
,
∀
p
⊂
0,1
,
2
,
…
,
m
-
1
,where(5)
B
j
s
=
b
n
∗
,
j
s
n
∗
+
b
n
∗
-
1
,
j
s
n
∗
-
1
+
⋯
+
b
1
,
j
s
+
b
0
,
j
,
j
=
1,2
,
…
,
m
.Assumption 5.
The sign ofb
n
∗
,
j is known for j
=
1,2
,
…
,
m.For adaptive actuator failure compensation, the proportional actuation scheme is(6)
v
j
=
s
i
g
n
b
n
∗
j
1
β
j
y
v
0
,
j
=
1,2
,
…
,
m
,where v
0 is a control signal generated by a backstepping design procedure to be given in Section 3.3.


### 3.1. Parameterized Model with Actuator Failures

To obtain a compact form of system (1) with actuator failures, we define(7)
k
1
,
r
=
∑
j
≠
j
1
,
j
2
,
…
,
j
p
s
i
g
n
b
n
∗
,
j
b
r
,
j
∗
ρ
j
,
r
=
0,1
,
…
,
n
∗
,
k
2
,
r
j
=
b
r
j
u
k
j
,
r
=
0,1
,
…
,
n
∗
,
j
=
j
1
,
j
2
,
…
,
j
p
,
0
,
r
=
0,1
,
…
,
n
∗
,
j
≠
j
1
,
j
2
,
…
,
j
p
.Applying (6)-(7) to (1), we have(8)
x
˙
i
=
x
i
+
1
+
φ
0
i
y
+
ϕ
i
x
+
a
T
φ
i
i
=
1,2
,
…
,
ρ
-
1
,
x
˙
ρ
=
x
ρ
+
1
+
φ
0
,
ρ
y
+
ϕ
ρ
x
+
a
T
φ
ρ
+
∑
j
=
1
m
k
2
,
n
∗
j
β
j
y
+
k
1
,
n
∗
v
0
⋮
x
˙
n
-
1
=
x
n
+
φ
0
,
n
-
1
y
+
ϕ
n
-
1
x
+
a
T
φ
n
-
1
+
∑
j
=
1
m
k
2,1
j
β
j
y
+
k
1,1
v
0
,
x
˙
n
=
φ
0
,
n
y
+
ϕ
n
x
+
a
T
φ
n
+
∑
j
=
1
m
k
2,0
j
β
j
y
+
k
1,0
v
0
,
y
=
x
1
.Let(9)
k
1
=
k
1
,
n
∗
,
…
,
k
1,0
T
∈
R
n
∗
+
1
,
k
2
,
r
=
k
2
,
r
1
,
k
2
,
r
2
,
…
,
k
2
,
r
m
T
∈
R
m
,
k
2
=
k
2
,
n
∗
,
…
,
k
2,0
T
∈
R
n
∗
+
1
m
,where r
=
0,1
,
…
,
n
∗, j
=
1,2
,
…
,
m, i
=
1,2
,
…
,
n. Substituting (9) into (8), we have(10)
x
˙
=
A
x
+
φ
0
y
+
ϕ
x
+
φ
a
y
a
+
Φ
k
2
y
k
2
+
0
ρ
-
1
×
1
k
1
v
0
,
y
=
e
1
T
x
,where(11)
x
=
x
1
,
x
2
,
…
,
x
n
T
,
A
=
0
1
0
⋯
0
0
0
1
⋯
0
⋮
⋮
⋮
⋮
⋮
0
0
0
⋯
1
0
0
0
⋯
0
,
e
1
=
1
0
⋮
0
0
T
,
φ
0
y
=
φ
0,1
y
,
φ
0,2
y
,
…
,
φ
0
,
n
y
T
,
φ
a
y
=
φ
1
y
,
φ
2
y
,
…
,
φ
n
y
T
,
Φ
k
2
y
=
0
ρ
-
1
×
n
∗
+
1
m
β
T
y
0
⋯
0
0
β
T
y
⋯
0
⋱
⋱
0
⋯
0
β
T
y
.We rewrite the system (10) as(12)
x
˙
=
A
x
+
φ
0
y
+
ϕ
x
+
F
y
,
t
,
v
0
T
θ
,
y
=
e
1
T
x
,where(13)
θ
=
k
1
T
,
k
2
T
,
a
T
T
∈
R
n
∗
+
1
×
m
+
1
+
q
,
Φ
y
,
t
=
Φ
k
2
y
,
t
,
φ
a
y
,
F
y
,
t
,
v
0
T
=
0
ρ
-
1
×
1
k
1
v
0
,
Φ
y
,
t
.Remark 6.
It is significant to point that the parametric model (12) is very similar to (
8.3
) and (
8.7
) in [26] and (
38
) in [9]. However, the system has nonlinearity terms ϕ
i
(
x
) which are functions of state vector. In other words, the design procedures in [9, 26] are not applicable for the system. In this paper, we will develop adaptive high-gain K-filters to deal with this problem.


### 3.2. State Observation and Switching Laws

Since the states of system (1) are not available, an observer is needed to provide auxiliary signals for handling the unmeasured states in control design. The conventional K-filters cannot deal with the nonlinearities depending on the system state. Motivated by [15, 16, 25], we develop adaptive high-gain K-filters which can be expressed as(14)
ξ
˙
=
A
0
L
ξ
+
φ
0
y
+
l
L
y
,
Ξ
˙
=
A
0
L
Ξ
+
Φ
y
,
t
,
r
=
0,1
,
…
,
n
∗
,
j
=
1,2
,
…
,
m
,
λ
.
=
A
0
L
λ
+
e
n
v
0
,
v
j
=
A
0
L
j
λ
,
j
=
0,1
,
…
,
n
∗
,
Ω
T
=
v
n
∗
,
…
,
v
0
,
Ξ
,where l
=
l
1
,
l
2
,
…
,
l
n
T is chosen such that A
0
=
A
-
l
e
1
T is Hurwitz, l
(
L
)
=
L
l
1
,
L
2
l
2
,
…
,
L
n
l
n
T, A
0
(
L
)
=
A
-
l
(
L
)
e
1
T, and L
≥
1. As a result, x
^
=
ξ
+
Ω
T
θ is an observer for x with the estimation error which satisfies(15)
ε
˙
=
A
0
L
ε
+
ϕ
x
.Remark 7.
SinceL
≥
1 and it varies with the system nonlinearities and actuator failures, we call it adaptive high-gain K-filters. But L is so complicated that we cannot solve an analytic value. Moreover, when actuator failures occur, the uncertainties will be also brought into the system. In the following, an adaptive controller is proposed which can tune the high-gain parameters online according to system nonlinearities and actuator failures.
Tuning mechanism and switching signalχ
(
t
): motivated by [15, 16], we consider the integral of e
(
t
) as(16)
χ
=
∫
t
0
t
0
+
T
e
t
d
t
,
χ
t
0
=
χ
0
,where(17)
e
t
=
y
t
-
y
r
t
.
The tuning steps are shown as follows.
Initialization. Set χ
0
=
0, L
0
=
1.Step 1.
ObtainS
=
a, where a is positive constant.Step 2.
Ifχ
≥
S, L
i
+
1
(
t
)
=
K
L
i
(
t
), where K
>
1, χ
=
0
→
i
=
i
+
1. Go to Step 1.
ElseL
i
+
1
(
t
)
=
L
i
(
t
), χ
=
0
→
i
=
i
+
1. Go to Step 2.Remark 8.
We briefly explain the idea of the tuning mechanism with the switching law. First,L
(
t
) is a monotonically nondecreasing function on the interval t
0
,
t
0
+
T by taking integral value of e
(
t
). Obviously, χ stops increasing only when e
(
t
)
=
0. Under this modified rule, once the system is stabilized, any integral of errors smaller than the prespecified value will not cause further switching. Also, from the switching logic, it is obvious that L
(
t
) is a piecewise constant function of time and increases stepwise with some time interval.


### 3.3. Adaptive Output Feedback Controller

To prepare for the backstepping procedure, we consider the equation for the outputy
=
x
1:(18)
y
˙
=
ω
0
+
ω
T
θ
+
ε
2
+
ϕ
1
=
k
1
,
n
∗
v
n
∗
,
2
+
ω
0
+
ω
-
T
θ
+
ε
2
+
ϕ
1
,where the regressor ω and truncated regressor ω
- are defined as(19)
ω
=
v
n
∗
,
2
,
v
n
∗
-
1,2
,
…
,
v
0,2
,
Φ
1
+
Ξ
2
T
,
ω
-
=
0
,
v
n
∗
-
1,2
,
…
,
v
0,2
,
Φ
1
+
Ξ
2
T
,
ω
0
=
φ
0,1
y
+
ξ
2
.Define a change of coordinates(20)
z
1
=
y
-
y
r
,
z
i
=
v
m
,
i
-
ρ
^
0
y
r
i
-
1
-
α
i
-
1
,
i
=
2
,
…
,
ρ
,where ρ
^
0 is an estimate of ρ
0
=
1
/
k
1
,
n
∗.Step 1. Consider (21)
z
1
=
y
-
y
r
,
α
-
1
=
-
c
1
z
1
-
d
1
+
L
ρ
4
z
1
-
ω
0
-
ω
-
T
θ
^
-
W
0
z
1
z
1
2
+
Δ
0
2
,
α
1
=
ρ
^
0
α
-
1
,
τ
1
=
10.1
c
m
00.1
c
m
⋯
0.1
c
m
0
T
ω
-
ρ
^
0
y
˙
r
+
α
-
1
×
10.1
c
m
00.1
c
m
⋯
0.1
c
m
0
T
z
1
,
ρ
^
˙
0
=
-
γ
s
g
n
k
1
,
n
∗
y
˙
r
+
α
-
1
z
1
,
W
0
=
f
L
ξ
+
Ω
1
T
θ
∗
+
⋯
+
Ω
n
T
θ
∗
,where α
1 is the first stabilizing function and τ
1 is the first tuning function, θ
∗, Δ
0 are design parameters, and f
(
L
)
≥
1 is an adaptive tuning function.Step
2. Consider(22)
z
2
=
v
m
,
2
-
ρ
^
0
y
˙
r
-
α
1
,
α
2
=
-
k
^
1
,
n
∗
z
1
-
c
2
z
2
-
d
2
+
L
ρ
4
∂
α
1
∂
y
2
z
2
+
y
.
r
+
∂
α
1
∂
ρ
^
0
ρ
^
˙
0
+
∂
α
1
∂
θ
^
T
τ
2
+
∂
α
1
∂
y
ω
0
+
ω
T
θ
^
+
∂
α
1
∂
ξ
A
0
L
ξ
+
l
L
y
+
φ
y
+
∂
α
1
∂
Ξ
A
0
L
Ξ
+
Φ
y
,
t
+
∂
α
1
∂
y
r
y
˙
r
+
l
L
2
v
n
∗
,
1
+
∑
j
=
1
m
+
1
∂
α
1
∂
λ
j
-
l
L
j
λ
1
+
λ
j
+
1
-
W
0
z
2
z
1
2
+
Δ
0
2
,
τ
2
=
τ
1
-
∂
α
1
∂
y
ω
z
2
.
Step
i, i
=
3
,
…
,
ρ
-
1. Consider (23)
z
i
=
v
m
,
i
-
ρ
^
0
y
r
i
-
1
-
α
i
-
1
,
α
i
=
-
z
i
-
1
-
c
i
z
i
-
d
i
+
L
ρ
4
∂
α
i
-
1
∂
y
2
z
i
+
y
r
i
-
1
+
∂
α
i
-
1
∂
ρ
^
0
ρ
^
˙
0
+
∂
α
i
-
1
∂
θ
^
T
τ
i
-
∑
j
=
2
i
-
1
∂
α
j
-
1
∂
θ
^
T
∂
α
j
-
1
∂
y
z
j
+
∂
α
i
-
1
∂
y
ω
0
+
ω
T
θ
^
+
∂
α
i
-
1
∂
ξ
A
0
L
ξ
+
l
L
y
+
φ
y
+
∂
α
i
-
1
∂
Ξ
A
0
L
Ξ
+
Φ
y
,
t
+
∑
j
=
1
i
-
1
∂
α
i
-
1
∂
y
r
j
-
1
y
r
j
+
l
L
i
v
n
∗
,
1
+
∑
j
=
1
m
-
1
+
i
∂
α
i
-
1
∂
λ
j
-
l
L
j
λ
1
+
λ
j
+
1
-
W
0
z
i
z
1
2
+
Δ
0
2
,
τ
i
=
τ
i
-
1
-
∂
α
i
-
1
∂
y
ω
z
i
-
1
.Step
ρ. Consider (24)
z
ρ
=
v
m
,
ρ
-
ρ
^
0
y
r
ρ
-
1
-
α
ρ
-
1
,
α
ρ
=
-
z
ρ
-
1
-
c
ρ
z
ρ
-
d
ρ
+
L
ρ
4
∂
α
ρ
-
1
∂
y
2
z
ρ
+
y
r
ρ
-
1
+
∂
α
ρ
-
1
∂
ρ
^
0
ρ
^
˙
0
+
∂
α
ρ
-
1
∂
θ
^
T
τ
ρ
+
∂
α
ρ
-
1
∂
y
ω
0
+
ω
T
θ
^
+
∂
α
ρ
-
1
∂
ξ
A
0
L
ξ
+
l
L
y
+
φ
y
+
∂
α
ρ
-
1
∂
Ξ
A
0
L
Ξ
+
Φ
y
,
t
+
∑
j
=
1
ρ
-
1
∂
α
ρ
-
1
∂
y
r
j
-
1
y
r
j
+
l
L
i
v
n
∗
,
1
+
∑
j
=
1
m
-
1
+
ρ
∂
α
i
-
1
∂
λ
j
-
l
L
j
λ
1
+
λ
j
+
1
-
∑
j
=
2
ρ
-
1
∂
α
j
-
1
∂
θ
^
T
∂
α
j
-
1
∂
y
z
ρ
-
W
0
z
ρ
z
1
2
+
Δ
0
2
,
τ
ρ
=
τ
ρ
-
1
-
∂
α
ρ
-
1
∂
y
ω
z
ρ
-
1
,where α
ρ is the ρth stabilizing function and τ
ρ is the ρth tuning function.Finally, the actual control signal and parameter adaptive laws are, respectively, designed as(25)
v
0
=
α
ρ
-
v
n
∗
,
ρ
+
1
+
ρ
^
0
y
r
ρ
,
(26)
θ
^
.
=
T
τ
ρ
,where T
=
T
T
>
0 is the adaptive gain.


## 3.1. Parameterized Model with Actuator Failures

To obtain a compact form of system (1) with actuator failures, we define(7)
k
1
,
r
=
∑
j
≠
j
1
,
j
2
,
…
,
j
p
s
i
g
n
b
n
∗
,
j
b
r
,
j
∗
ρ
j
,
r
=
0,1
,
…
,
n
∗
,
k
2
,
r
j
=
b
r
j
u
k
j
,
r
=
0,1
,
…
,
n
∗
,
j
=
j
1
,
j
2
,
…
,
j
p
,
0
,
r
=
0,1
,
…
,
n
∗
,
j
≠
j
1
,
j
2
,
…
,
j
p
.Applying (6)-(7) to (1), we have(8)
x
˙
i
=
x
i
+
1
+
φ
0
i
y
+
ϕ
i
x
+
a
T
φ
i
i
=
1,2
,
…
,
ρ
-
1
,
x
˙
ρ
=
x
ρ
+
1
+
φ
0
,
ρ
y
+
ϕ
ρ
x
+
a
T
φ
ρ
+
∑
j
=
1
m
k
2
,
n
∗
j
β
j
y
+
k
1
,
n
∗
v
0
⋮
x
˙
n
-
1
=
x
n
+
φ
0
,
n
-
1
y
+
ϕ
n
-
1
x
+
a
T
φ
n
-
1
+
∑
j
=
1
m
k
2,1
j
β
j
y
+
k
1,1
v
0
,
x
˙
n
=
φ
0
,
n
y
+
ϕ
n
x
+
a
T
φ
n
+
∑
j
=
1
m
k
2,0
j
β
j
y
+
k
1,0
v
0
,
y
=
x
1
.Let(9)
k
1
=
k
1
,
n
∗
,
…
,
k
1,0
T
∈
R
n
∗
+
1
,
k
2
,
r
=
k
2
,
r
1
,
k
2
,
r
2
,
…
,
k
2
,
r
m
T
∈
R
m
,
k
2
=
k
2
,
n
∗
,
…
,
k
2,0
T
∈
R
n
∗
+
1
m
,where r
=
0,1
,
…
,
n
∗, j
=
1,2
,
…
,
m, i
=
1,2
,
…
,
n. Substituting (9) into (8), we have(10)
x
˙
=
A
x
+
φ
0
y
+
ϕ
x
+
φ
a
y
a
+
Φ
k
2
y
k
2
+
0
ρ
-
1
×
1
k
1
v
0
,
y
=
e
1
T
x
,where(11)
x
=
x
1
,
x
2
,
…
,
x
n
T
,
A
=
0
1
0
⋯
0
0
0
1
⋯
0
⋮
⋮
⋮
⋮
⋮
0
0
0
⋯
1
0
0
0
⋯
0
,
e
1
=
1
0
⋮
0
0
T
,
φ
0
y
=
φ
0,1
y
,
φ
0,2
y
,
…
,
φ
0
,
n
y
T
,
φ
a
y
=
φ
1
y
,
φ
2
y
,
…
,
φ
n
y
T
,
Φ
k
2
y
=
0
ρ
-
1
×
n
∗
+
1
m
β
T
y
0
⋯
0
0
β
T
y
⋯
0
⋱
⋱
0
⋯
0
β
T
y
.We rewrite the system (10) as(12)
x
˙
=
A
x
+
φ
0
y
+
ϕ
x
+
F
y
,
t
,
v
0
T
θ
,
y
=
e
1
T
x
,where(13)
θ
=
k
1
T
,
k
2
T
,
a
T
T
∈
R
n
∗
+
1
×
m
+
1
+
q
,
Φ
y
,
t
=
Φ
k
2
y
,
t
,
φ
a
y
,
F
y
,
t
,
v
0
T
=
0
ρ
-
1
×
1
k
1
v
0
,
Φ
y
,
t
.Remark 6.
It is significant to point that the parametric model (12) is very similar to (
8.3
) and (
8.7
) in [26] and (
38
) in [9]. However, the system has nonlinearity terms ϕ
i
(
x
) which are functions of state vector. In other words, the design procedures in [9, 26] are not applicable for the system. In this paper, we will develop adaptive high-gain K-filters to deal with this problem.


## 3.2. State Observation and Switching Laws

Since the states of system (1) are not available, an observer is needed to provide auxiliary signals for handling the unmeasured states in control design. The conventional K-filters cannot deal with the nonlinearities depending on the system state. Motivated by [15, 16, 25], we develop adaptive high-gain K-filters which can be expressed as(14)
ξ
˙
=
A
0
L
ξ
+
φ
0
y
+
l
L
y
,
Ξ
˙
=
A
0
L
Ξ
+
Φ
y
,
t
,
r
=
0,1
,
…
,
n
∗
,
j
=
1,2
,
…
,
m
,
λ
.
=
A
0
L
λ
+
e
n
v
0
,
v
j
=
A
0
L
j
λ
,
j
=
0,1
,
…
,
n
∗
,
Ω
T
=
v
n
∗
,
…
,
v
0
,
Ξ
,where l
=
l
1
,
l
2
,
…
,
l
n
T is chosen such that A
0
=
A
-
l
e
1
T is Hurwitz, l
(
L
)
=
L
l
1
,
L
2
l
2
,
…
,
L
n
l
n
T, A
0
(
L
)
=
A
-
l
(
L
)
e
1
T, and L
≥
1. As a result, x
^
=
ξ
+
Ω
T
θ is an observer for x with the estimation error which satisfies(15)
ε
˙
=
A
0
L
ε
+
ϕ
x
.Remark 7.
SinceL
≥
1 and it varies with the system nonlinearities and actuator failures, we call it adaptive high-gain K-filters. But L is so complicated that we cannot solve an analytic value. Moreover, when actuator failures occur, the uncertainties will be also brought into the system. In the following, an adaptive controller is proposed which can tune the high-gain parameters online according to system nonlinearities and actuator failures.
Tuning mechanism and switching signalχ
(
t
): motivated by [15, 16], we consider the integral of e
(
t
) as(16)
χ
=
∫
t
0
t
0
+
T
e
t
d
t
,
χ
t
0
=
χ
0
,where(17)
e
t
=
y
t
-
y
r
t
.
The tuning steps are shown as follows.
Initialization. Set χ
0
=
0, L
0
=
1.Step 1.
ObtainS
=
a, where a is positive constant.Step 2.
Ifχ
≥
S, L
i
+
1
(
t
)
=
K
L
i
(
t
), where K
>
1, χ
=
0
→
i
=
i
+
1. Go to Step 1.
ElseL
i
+
1
(
t
)
=
L
i
(
t
), χ
=
0
→
i
=
i
+
1. Go to Step 2.Remark 8.
We briefly explain the idea of the tuning mechanism with the switching law. First,L
(
t
) is a monotonically nondecreasing function on the interval t
0
,
t
0
+
T by taking integral value of e
(
t
). Obviously, χ stops increasing only when e
(
t
)
=
0. Under this modified rule, once the system is stabilized, any integral of errors smaller than the prespecified value will not cause further switching. Also, from the switching logic, it is obvious that L
(
t
) is a piecewise constant function of time and increases stepwise with some time interval.


## 3.3. Adaptive Output Feedback Controller

To prepare for the backstepping procedure, we consider the equation for the outputy
=
x
1:(18)
y
˙
=
ω
0
+
ω
T
θ
+
ε
2
+
ϕ
1
=
k
1
,
n
∗
v
n
∗
,
2
+
ω
0
+
ω
-
T
θ
+
ε
2
+
ϕ
1
,where the regressor ω and truncated regressor ω
- are defined as(19)
ω
=
v
n
∗
,
2
,
v
n
∗
-
1,2
,
…
,
v
0,2
,
Φ
1
+
Ξ
2
T
,
ω
-
=
0
,
v
n
∗
-
1,2
,
…
,
v
0,2
,
Φ
1
+
Ξ
2
T
,
ω
0
=
φ
0,1
y
+
ξ
2
.Define a change of coordinates(20)
z
1
=
y
-
y
r
,
z
i
=
v
m
,
i
-
ρ
^
0
y
r
i
-
1
-
α
i
-
1
,
i
=
2
,
…
,
ρ
,where ρ
^
0 is an estimate of ρ
0
=
1
/
k
1
,
n
∗.Step 1. Consider (21)
z
1
=
y
-
y
r
,
α
-
1
=
-
c
1
z
1
-
d
1
+
L
ρ
4
z
1
-
ω
0
-
ω
-
T
θ
^
-
W
0
z
1
z
1
2
+
Δ
0
2
,
α
1
=
ρ
^
0
α
-
1
,
τ
1
=
10.1
c
m
00.1
c
m
⋯
0.1
c
m
0
T
ω
-
ρ
^
0
y
˙
r
+
α
-
1
×
10.1
c
m
00.1
c
m
⋯
0.1
c
m
0
T
z
1
,
ρ
^
˙
0
=
-
γ
s
g
n
k
1
,
n
∗
y
˙
r
+
α
-
1
z
1
,
W
0
=
f
L
ξ
+
Ω
1
T
θ
∗
+
⋯
+
Ω
n
T
θ
∗
,where α
1 is the first stabilizing function and τ
1 is the first tuning function, θ
∗, Δ
0 are design parameters, and f
(
L
)
≥
1 is an adaptive tuning function.Step
2. Consider(22)
z
2
=
v
m
,
2
-
ρ
^
0
y
˙
r
-
α
1
,
α
2
=
-
k
^
1
,
n
∗
z
1
-
c
2
z
2
-
d
2
+
L
ρ
4
∂
α
1
∂
y
2
z
2
+
y
.
r
+
∂
α
1
∂
ρ
^
0
ρ
^
˙
0
+
∂
α
1
∂
θ
^
T
τ
2
+
∂
α
1
∂
y
ω
0
+
ω
T
θ
^
+
∂
α
1
∂
ξ
A
0
L
ξ
+
l
L
y
+
φ
y
+
∂
α
1
∂
Ξ
A
0
L
Ξ
+
Φ
y
,
t
+
∂
α
1
∂
y
r
y
˙
r
+
l
L
2
v
n
∗
,
1
+
∑
j
=
1
m
+
1
∂
α
1
∂
λ
j
-
l
L
j
λ
1
+
λ
j
+
1
-
W
0
z
2
z
1
2
+
Δ
0
2
,
τ
2
=
τ
1
-
∂
α
1
∂
y
ω
z
2
.
Step
i, i
=
3
,
…
,
ρ
-
1. Consider (23)
z
i
=
v
m
,
i
-
ρ
^
0
y
r
i
-
1
-
α
i
-
1
,
α
i
=
-
z
i
-
1
-
c
i
z
i
-
d
i
+
L
ρ
4
∂
α
i
-
1
∂
y
2
z
i
+
y
r
i
-
1
+
∂
α
i
-
1
∂
ρ
^
0
ρ
^
˙
0
+
∂
α
i
-
1
∂
θ
^
T
τ
i
-
∑
j
=
2
i
-
1
∂
α
j
-
1
∂
θ
^
T
∂
α
j
-
1
∂
y
z
j
+
∂
α
i
-
1
∂
y
ω
0
+
ω
T
θ
^
+
∂
α
i
-
1
∂
ξ
A
0
L
ξ
+
l
L
y
+
φ
y
+
∂
α
i
-
1
∂
Ξ
A
0
L
Ξ
+
Φ
y
,
t
+
∑
j
=
1
i
-
1
∂
α
i
-
1
∂
y
r
j
-
1
y
r
j
+
l
L
i
v
n
∗
,
1
+
∑
j
=
1
m
-
1
+
i
∂
α
i
-
1
∂
λ
j
-
l
L
j
λ
1
+
λ
j
+
1
-
W
0
z
i
z
1
2
+
Δ
0
2
,
τ
i
=
τ
i
-
1
-
∂
α
i
-
1
∂
y
ω
z
i
-
1
.Step
ρ. Consider (24)
z
ρ
=
v
m
,
ρ
-
ρ
^
0
y
r
ρ
-
1
-
α
ρ
-
1
,
α
ρ
=
-
z
ρ
-
1
-
c
ρ
z
ρ
-
d
ρ
+
L
ρ
4
∂
α
ρ
-
1
∂
y
2
z
ρ
+
y
r
ρ
-
1
+
∂
α
ρ
-
1
∂
ρ
^
0
ρ
^
˙
0
+
∂
α
ρ
-
1
∂
θ
^
T
τ
ρ
+
∂
α
ρ
-
1
∂
y
ω
0
+
ω
T
θ
^
+
∂
α
ρ
-
1
∂
ξ
A
0
L
ξ
+
l
L
y
+
φ
y
+
∂
α
ρ
-
1
∂
Ξ
A
0
L
Ξ
+
Φ
y
,
t
+
∑
j
=
1
ρ
-
1
∂
α
ρ
-
1
∂
y
r
j
-
1
y
r
j
+
l
L
i
v
n
∗
,
1
+
∑
j
=
1
m
-
1
+
ρ
∂
α
i
-
1
∂
λ
j
-
l
L
j
λ
1
+
λ
j
+
1
-
∑
j
=
2
ρ
-
1
∂
α
j
-
1
∂
θ
^
T
∂
α
j
-
1
∂
y
z
ρ
-
W
0
z
ρ
z
1
2
+
Δ
0
2
,
τ
ρ
=
τ
ρ
-
1
-
∂
α
ρ
-
1
∂
y
ω
z
ρ
-
1
,where α
ρ is the ρth stabilizing function and τ
ρ is the ρth tuning function.Finally, the actual control signal and parameter adaptive laws are, respectively, designed as(25)
v
0
=
α
ρ
-
v
n
∗
,
ρ
+
1
+
ρ
^
0
y
r
ρ
,
(26)
θ
^
.
=
T
τ
ρ
,where T
=
T
T
>
0 is the adaptive gain.


## 4. Stability Analysis

To prepare for the stability analysis, we rewrite the error system as(27)
z
˙
=
A
z
z
,
t
+
A
c
z
,
t
z
+
W
ε
z
,
t
ε
2
+
ϕ
1
+
W
θ
z
,
t
T
θ
~
-
k
1
,
n
∗
y
˙
r
+
α
-
1
e
1
ρ
~
0
-
Γ
z
W
0
,where the system matrices A
z
(
z
,
t
), W
ε
(
z
,
t
), W
θ
(
z
,
t
), W
θ
(
z
,
t
), A
c
(
z
,
t
), Γ
z are given by(28)
A
z
z
,
t
=
-
c
1
-
d
1
k
^
1
,
n
∗
0
⋯
⋯
0
k
^
1
,
n
∗
-
c
2
-
d
2
∂
α
1
∂
y
2
1
+
σ
23
σ
24
⋯
σ
2
ρ
⋮
-
1
-
σ
23
⋱
⋱
⋱
⋮
⋮
-
σ
24
⋱
⋱
⋱
σ
ρ
-
2
ρ
⋮
⋮
⋱
⋱
⋱
1
+
σ
ρ
-
1
,
ρ
0
-
σ
2
ρ
⋯
-
σ
ρ
-
2
,
ρ
-
1
-
σ
ρ
-
1
,
ρ
-
c
ρ
-
d
ρ
∂
α
ρ
-
1
∂
y
2
,
W
ε
z
,
t
=
1
-
∂
α
1
∂
y
-
∂
α
2
∂
y
-
⋯
-
∂
α
ρ
-
1
∂
y
T
∈
R
ρ
,
W
θ
z
,
t
=
W
ε
z
,
t
ω
T
-
ρ
^
0
y
˙
r
+
α
-
1
e
1
e
1
T
∈
R
ρ
×
ρ
,
A
c
z
,
t
=
-
L
ρ
4
0
0
⋯
0
0
-
L
ρ
4
∂
α
1
∂
y
2
0
⋯
0
0
⋮
⋱
⋮
⋮
0
0
⋯
-
L
ρ
4
∂
α
ρ
-
1
∂
y
2
,
Γ
z
=
z
1
z
1
2
+
Δ
0
2
,
z
2
z
1
2
+
Δ
0
2
,
…
,
z
ρ
z
1
2
+
Δ
0
2
T
.Let(29)
E
L
=
d
i
a
g
1
,
1
L
⋯
1
L
n
-
1
.The candidate Lyapunov function for the closed-loop system is chosen as(30)
V
=
1
2
z
T
z
+
1
2
θ
~
T
-
1
θ
~
T
+
k
1
,
n
∗
2
γ
ρ
~
2
+
ε
T
P
L
ε
,where (31)
A
0
T
P
+
P
A
0
=
-
I
,
P
L
=
E
L
P
E
L
.The proposed adaptive scheme has the following properties.Theorem 9.
The adaptive output feedback control scheme consisting of the controller (25) and the filters (14) along with the parameter update laws (26) applied to the system (1) based on Assumptions 1–5 ensures global boundedness of all closed-loop signals and desired output tracking performance: (32)
∫
t
1
t
2
y
t
-
y
r
t
2
d
t
≤
λ
c
0
t
2
-
t
1
+
γ
0
,for any t
2
>
t
1
≥
0, where λ
>
0, γ
0
>
0, c
0
>
0 are design parameters. The tracking error can be made as small as desired by choosing sufficiently large c
0 and L.Proof.
For each time interval(
t
k
,
t
k
+
1
), k
=
0,1
,
2
,
…
,
q, we have a Lyapunov function V defined in (30). Recalling (27), the derivative of V is(33)
V
˙
=
1
2
z
T
A
z
+
A
z
T
z
+
z
T
A
c
z
,
t
z
+
z
T
W
ε
ε
2
+
z
T
W
ϕ
1
+
z
T
W
θ
T
θ
~
+
2
ε
T
E
L
P
E
L
ϕ
-
θ
~
T
W
θ
z
-
z
T
k
1
,
n
∗
y
˙
r
+
α
-
1
e
1
ρ
~
0
+
ρ
~
0
k
1
,
n
∗
y
˙
r
+
α
-
1
e
1
T
z
+
ε
T
A
0
T
L
P
L
+
P
L
A
0
L
ε
-
z
2
z
1
2
+
Δ
0
2
f
L
W
0
.Then, observing that(34)
2
ε
T
E
L
P
E
L
ϕ
=
2
L
ε
T
E
L
P
ϕ
1
L
ϕ
2
L
2
⋮
ϕ
n
L
n
≤
2
L
ε
E
L
P
φ
,where φ
=
φ
1
,
φ
2
,
…
,
φ
n
T
=
ϕ
1
/
L
,
ϕ
2
/
L
2
,
…
,
ϕ
n
/
L
n
T and from Assumption 1, we estimate the elements of φ
(35)
ϕ
i
L
i
≤
c
L
i
x
^
1
+
⋯
+
x
^
i
+
ε
1
+
⋯
+
ε
i
≤
c
i
x
^
+
ε
,where c
i
=
c
n
/
L
i.
Hence,(36)
2
L
ε
E
L
P
φ
=
2
L
E
L
P
ε
φ
≤
2
C
ε
x
^
+
ε
,where C is a positive constant depending on x
^, P, E
L, c, n, and L.
Note that(37)
z
T
A
c
z
,
t
z
+
z
T
W
ϕ
1
=
ϕ
1
2
L
-
L
ρ
2
z
1
-
1
L
ρ
ϕ
1
2
-
∑
i
=
2
ρ
L
ρ
2
∂
α
i
-
1
∂
y
z
i
+
1
L
ρ
ϕ
1
2
≤
c
2
L
y
2
≤
2
c
2
L
y
r
2
+
z
1
2
.
Substituting (34) and (36) into (33), we have(38)
V
˙
≤
-
∑
i
=
1
ρ
c
i
z
i
2
+
2
c
2
L
z
1
2
-
∑
i
=
1
ρ
d
i
∂
α
i
-
1
∂
y
z
i
2
-
∑
i
=
1
ρ
z
i
∂
α
i
-
1
∂
y
ε
2
-
L
ε
2
+
2
C
ε
x
^
+
ε
+
2
c
2
L
y
r
2
-
z
2
z
1
2
+
Δ
0
2
f
L
≤
-
∑
i
=
1
ρ
c
i
z
i
2
+
2
c
2
L
z
1
2
-
z
2
z
1
2
+
Δ
0
2
f
L
W
0
+
x
^
2
+
2
c
2
L
y
r
2
-
L
-
2
C
-
4
C
2
-
∑
i
=
1
ρ
1
4
d
i
ε
2
.With the choice of L
-
2
C
-
4
C
2
-
∑
i
=
1
ρ
(
1
/
4
d
i
)
>
0, c
1
-
2
c
2
>
0, and f
(
L
)
≥
(
(
z
1
2
+
Δ
0
2
)
/
z
2
)
×
(
θ
1
/
θ
∗
),(39)
V
˙
≤
-
∑
i
=
1
ρ
c
i
z
i
2
+
2
c
2
L
z
1
2
+
2
c
2
L
y
r
2
.Let (40)
λ
=
2
c
2
L
y
r
2
.
It should be noted thatλ can be made as small as desired by choosing sufficiently larger L. Recalling (39), we can obtain(41)
V
˙
≤
-
c
0
z
1
2
-
∑
i
=
2
ρ
c
i
z
i
2
+
λ
,where c
1
′
=
c
1
-
2
c
2.
Starting from the first interval, we conclude that(
t
)
∈
L
∞
∀
t
∈
t
0
,
t
1, so that z, θ
^, and ε are all bounded ∀
t
∈
t
0
,
t
1. From (14) and (25), it follows that x
^ is bounded. At time t
=
t
1, there occur p
1 actuator failures, which results in the abrupt change of the parameters. Since the change of values is finite, we have(42)
V
t
1
+
=
V
t
1
-
+
V
-
1
,where V
-
1 is bounded.
Therefore, it can be concluded from (41) that (
t
)
∈
L
∞
∀
t
∈
t
1
,
t
2. By repeating the argument above, the boundedness of all the signals is proved for the time interval t
1
,
t
2. Continuing in the same manner, finally we have that ∀
t
∈
t
q
,
∞, V
(
t
)
∈
L
∞ and so are the closed-loop signals.
To prove tracking performance, consider the last time intervalt
q
,
∞. From (41), we see that V
˙
≤
-
c
0
z
1
2
-
∑
i
=
2
ρ
c
i
z
i
2
+
λ. In particular, V
˙
≤
-
c
0
z
1
2
+
λ. Integrating both sides from t
=
t
1 to t
=
t
2, where t
2
>
t
1
≥
0, we have ∫
t
1
t
2
(
y
(
t
)
-
y
r
(
t
)
)
2
d
t
≤
(
λ
/
c
0
)
(
t
2
-
t
1
)
+
V
(
t
2
)
-
V
(
t
1
). Since V
(
t
)
∈
L
∞, there exists γ
0
>
0 such that the desired output tracking given by (32) is achieved. This completes the proof.


## 5. Examples and Simulations

In this section, we present two examples to illustrate the applications of Theorem9.Example 1.
The nonlinear longitudinal dynamics of the twin otter aircraft [1, 7, 27] is used for our actuator failure compensation study. Choosing the velocity, angle of attack, pitch angle, and pitch rate as the states x
1, x
2, x
3, and x
4 and the elevator angles of an augmented two-piece elevator as the inputs u
1, u
2, the system can be modeled as(43)
x
˙
1
=
c
1
T
φ
0
x
2
x
1
2
+
φ
1
x
c
o
s
x
2
+
c
2
T
φ
0
x
2
x
2
2
+
φ
2
x
s
i
n
x
2
+
d
1
g
1
x
u
1
+
d
2
g
1
x
u
2
,
x
˙
2
=
x
4
-
c
1
T
φ
0
x
2
x
1
2
+
φ
1
x
1
x
1
s
i
n
x
2
+
c
2
T
φ
0
x
2
x
2
2
+
φ
2
x
1
x
1
c
o
s
x
2
+
d
1
g
2
x
u
1
+
d
2
g
2
x
u
2
,
x
˙
3
=
x
4
,
x
˙
4
=
θ
T
ϕ
x
+
b
1
x
1
2
u
1
+
b
2
x
1
2
u
2
,where(44)
φ
0
x
2
=
x
2
,
x
2
2
,
1
T
,
φ
1
x
=
p
11
+
p
12
x
4
x
1
2
-
p
0
s
i
n
x
3
,
φ
2
x
=
p
21
+
p
22
x
4
x
1
2
+
p
0
c
o
s
x
3
,
g
1
x
=
a
1
x
1
2
c
o
s
x
2
+
a
2
x
1
2
s
i
n
x
2
,
g
2
x
=
-
a
1
x
1
s
i
n
x
2
+
a
2
x
1
c
o
s
x
2
,
ϕ
x
=
x
1
2
x
2
,
x
1
2
x
2
2
,
x
1
2
,
x
1
2
x
4
.For this application,x
3 is considered as the output; that is, y
=
x
3, x
1, and the velocity is measured while x
2, x
4 are unmeasured. Apparently, x
3
,
x
4
T subsystem can be described as(45)
x
˙
3
=
x
4
,
x
˙
4
=
θ
T
ϕ
x
+
b
T
x
1
2
u
,with some unknown parameters b
∈
R
2 and u
=
u
1
,
u
2
T. The x
1
,
x
2
T subsystem constructs the zero dynamics, which is input-to-state stability [1, 7]. The unmeasured state x
4 exhibits the appearance of a linear form ϕ
(
x
), which satisfies Assumption 1. Taking advantage of this property, we design adaptive high-gain K-filters to estimate x
4.The control objective is to design an adaptive scheme to control the elevator angles such that the pitch angley
=
x
3 tracks a reference signal y
r as close as desired even if one piece of the elevator is stuck at an unknown angle at an unknown time instant. For simulation, we consider two actuator failure cases.Case 1.
We consider the case whereu
1 fails at the 50th second. Thus, u
1 undergoes a TLOE type of failure: (46)
u
1
t
=
v
1
t
,
u
2
t
=
ρ
2
v
2
+
u
k
2
,where ρ
2
=
1, u
k
2
=
0, and t
∈
0,100; ρ
2
=
0, u
k
2
=
-
0.08, and t
∈
100
,
∞.
By Theorem9, we can obtain the actual control law, the high-gain K-filters, and the update laws. The initial conditions are set as(47)
c
1
=
0.01
,
c
2
=
0.01
,
d
1
=
1
,
d
2
=
1
,
l
=
6,8
T
,
θ
^
0
=
18,0
,
0
T
,
x
0
=
72,0.00,0.001,0.05
T
,
T
=
d
i
a
g
5,1
,
1
∗
50
.
The switching law parameters are chosen as(48)
T
=
1
,
a
=
0.000003
,
K
=
1.2
.
The simulation results including outputy
(
t
), reference output y
r
(
t
), and tracking error e
(
t
) are shown in Figure 1, the control inputs u
1, u
2 are shown in Figure 2, the adaptive parameter L and unmeasured state x
4 are shown in Figure 3, and the zero dynamics x
1 and x
2 are shown in Figure 4. The system responses are as expected. When one of the actuators fails, there is a transient response in tracking errors. But as time goes on, the tracking errors become smaller and ultimately vanish. The adaptive controller can tune its parameters according to the actuator failures, while the adaptive high-gain K-filters also change their value if necessary. Furthermore, it is easily seen that the unmeasured state x
4 and the zero dynamics x
1, x
2 are stable even though there is an actuator failing during operation.Figure 1
Plant outputy, reference y
r, and error e
(
t
).
(a)
(b)Figure 2
Plant inputsu
1, u
2.Figure 3
The adaptive parameterL and the unmeasured state x
4.
(a)
(b)Figure 4
The zero dynamicsx
1 and x
2.
(a)
(b)Case 2.
We consider the case whereu
2 (TLOE) fails at the 100th second and u
1 (PLOE) loses 50% of its effectiveness from t
=
160 s. That is,(49)
u
1
t
=
ρ
1
v
1
,
u
2
t
=
ρ
2
v
2
+
u
k
2
,where(50)
ρ
1
=
1
,
t
∈
0,160
;
ρ
1
=
0.5
,
t
∈
160
,
∞
,
ρ
2
=
1
,
t
∈
0,100
;
ρ
2
=
0
,
u
k
2
=
-
0.08
,
t
∈
100
,
∞
.
The other parameters are the same as those in Case1.
The simulation results including outputy
(
t
), reference output y
r
(
t
), and tracking error e
(
t
) are shown in Figure 5, the control inputs u
1, u
2 are shown in Figure 6, the adaptive parameter L and unmeasured state x
4 are shown in Figure 7, and the zero dynamics x
1 and x
2 are shown in Figure 8. The system responses are as expected. When one of the actuators fails, there is a transient response in tracking errors. But as time goes on, the tracking errors become smaller and ultimately vanish. The adaptive controller can tune its parameters according to the actuator failures, while the adaptive high-gain K-filters also change their value if necessary. Furthermore, it is easily seen that the unmeasured state x
4 and the zero dynamics x
1
,
x
2 are stable even though there is an actuator failing during operation.Figure 5
Plant outputy, reference y
r, and error e
(
t
).
(a)
(b)Figure 6
Plant inputsu
1, u
2.Figure 7
The adaptive parameterL and the unmeasured state x
4.
(a)
(b)Figure 8
The zero dynamicsx
1 and x
2.
(a)
(b)Example 2.
A rigid-body longitudinal model of a hypersonic aircraft cruising at a velocity of 15 Mach and at an altitude of 110,000 ft [1, 11] is considered:(51)
x
˙
1
=
x
2
+
f
1
x
3
,
y
,
x
˙
2
=
b
1
u
1
+
b
2
u
2
+
f
2
x
3
,
y
,
x
˙
3
=
f
3
x
3
,
y
,where(52)
f
1
x
3
,
y
=
θ
1
y
+
θ
2
s
i
n
y
+
θ
3
y
2
s
i
n
y
+
θ
4
c
o
s
x
3
,
f
2
x
2
,
y
=
θ
5
y
2
+
θ
6
y
+
θ
7
y
2
x
2
+
θ
8
y
x
2
+
θ
9
x
2
+
θ
11
,
f
3
x
3
,
y
=
θ
10
c
o
s
x
3
-
θ
1
y
+
θ
2
s
i
n
y
+
θ
3
y
2
s
i
n
y
.
We rewrite the plant (51) as(53)
x
˙
1
=
x
2
+
ϕ
1
x
3
+
φ
1
y
,
x
˙
2
=
ϕ
2
y
x
2
+
φ
2
y
+
b
1
u
1
+
b
2
u
2
,
x
˙
3
=
θ
10
c
o
s
x
3
-
φ
1
y
,
y
=
x
1
,where x
1 is the angle of attack, x
2 is the pitch rate and unmeasured, x
3 is the flight-path angle and unmeasured, and u
1
(
t
) and u
2
(
t
) are the elevator segment deflection angles. b
1 and b
2 are unknown constants with known signs, and ϕ
1, ϕ
2, φ
1, and φ
2 are known functions given by(54)
ϕ
1
x
3
=
θ
4
c
o
s
x
3
,
ϕ
2
y
=
θ
7
y
2
+
θ
8
y
+
θ
9
,
φ
1
y
=
θ
1
y
+
θ
2
s
i
n
y
+
θ
3
y
2
s
i
n
y
,
φ
2
y
=
θ
5
y
2
+
θ
6
y
+
θ
11
.
The control objective for actuator failure compensation of aircraft model is the angle of attackx
1 that tracks a reference signal y
r with a sufficiently small error, while the closed-loop system is stabilized in the presence of an actuator failure in either one of the elevator segments. Since y
=
x
1 is the output, equation x
˙
3
=
θ
10
c
o
s
(
x
3
)
-
φ
1
(
y
) constructs the zero dynamics. For simulation, we also consider two actuator failure cases.Case 1.
We consider the case whereu
2 (TLOE) fails at the 50th second. That is,(55)
u
1
t
=
v
1
t
,
u
2
t
=
ρ
2
v
2
+
u
k
2
,where ρ
2
=
1, u
k
2
=
0, and t
∈
0,50; ρ
2
=
0, u
k
2
=
0.1, and t
∈
50
,
∞. From Theorem 9, we can obtain the actual control law, the high-gain K-filters, and the update laws. The simulation parameters are as follows:(56)
c
1
=
0.001
,
c
2
=
0.001
,
d
1
=
1
,
d
2
=
15
,
l
=
6,8
T
,
θ
^
0
=
0,0
,
0
T
,
x
0
=
0,0
,
1.57
T
,
T
=
d
i
a
g
100,10,10
,
T
=
10
,
a
=
0.004
,
K
=
1.2
.
The simulation results including outputy
(
t
), reference output y
r
(
t
), and tracking error e
(
t
) are shown in Figure 9, the control inputs u
1, u
2 are shown in Figure 10, the adaptive parameter L is shown in Figure 11, and the adaptive parameter L is shown in Figure 12. The system responses are as expected. When one of the actuators fails, there is a transient response in tracking errors. But as time goes on, the tracking errors become smaller and ultimately vanish. The adaptive controller can tune its parameters according to the actuator failures, while the adaptive high-gain K-filters do not change their value. Furthermore, it is easily seen that the unmeasured state x
2 is stable even though there is an actuator failing during operation.Figure 9
Plant outputy, reference y
r, and error e
(
t
).
(a)
(b)Figure 10
Plant inputsu
1, u
2.Figure 11
The adaptive parameterL and the unmeasured state x
4.Figure 12
The zero dynamicsx
1, x
2.Case 2.
We consider the case whereu
1 (TLOE) fails at the 100th second and u
2 (PLOE) loses 40% of its effectiveness from t
=
150 s:(57)
u
1
t
=
ρ
1
v
1
t
+
u
k
1
,
u
2
t
=
ρ
2
v
2
t
,where(58)
ρ
1
=
1
,
u
k
1
=
0
,
t
∈
0,100
;
ρ
1
=
0
,
u
k
1
=
-
0.1
,
t
∈
100
,
∞
,
ρ
2
=
1
,
t
∈
0,150
;
ρ
2
=
0.6
,
t
∈
150
,
∞
.
The other parameters are the same as those in Case1.
The simulation results including outputy
(
t
), reference output y
r
(
t
), and tracking error e
(
t
) are shown in Figure 13, the control inputs u
1, u
2 are shown in Figure 14, the adaptive parameter L is shown in Figure 15, and the adaptive parameter L is shown in Figure 16. The system responses are as expected. When one of the actuators fails, there is a transient response in tracking errors. But as time goes on, the tracking errors become smaller and ultimately vanish. The adaptive controller can tune its parameters according to the actuator failures, while the adaptive high-gain K-filters do not change their value. Furthermore, it is easily seen that the unmeasured state x
2 is stable even though there is an actuator failing during operation.Figure 13
Plant outputy, reference y
r, and error e
(
t
).
(a)
(b)Figure 14
Plant inputsu
1, u
2.Figure 15
The adaptive parameterL and the unmeasured state x
4.Figure 16
The zero dynamicsx
1, x
2.


## 6. Conclusions

In this paper, adaptive high-gain K-filters and adaptive high-gain controller are applied to handle a class of nonlinear systems uncertainties in the presence of uncertain actuator failures. The adaptive high-gain K-filters can suppress the nonlinearities while the adaptive controller guarantees the closed-loop signals stability and small tracking errors. Simulation results verify the effectiveness of the adaptive actuator failure compensation for desired performance.



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*Source: 357078-2015-05-21.xml*