3.1. State Estimation
Control signals
u
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 that
κ
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 of
A
, 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
.
Let
x
~
=
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 coordinates
x
-
=
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 that
l
≥
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
.
Since
P
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
.
3.3. 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 output
y
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.
Differentiating
z
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 functions
κ
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
.
Differentiating
V
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 of
z
2
along with (31) is
(41)
z
˙
2
=
ν
m
,
3
-
q
2
l
2
ν
m
,
1
-
α
˙
1
.
Define the function
β
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
.
Define
z
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 of
z
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 of
V
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 of
z
i
along with (31) is
(47)
z
˙
i
=
ν
m
,
i
-
q
i
l
i
ν
m
,
1
-
α
˙
i
-
1
.
Choose stabilizing function
α
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 of
V
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
.
Define
z
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 of
V
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 time
t
,
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 Lemma 4, 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 matrix
Γ
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.