# Robust Fuzzy Control for Fractional-Order Uncertain Hydroturbine Regulating System with Random Disturbances

Authors: Fengjiao Wu; Guitao Zhang; Zhengzhong Wang
Journal: Mathematical Problems in Engineering
 (2016)

---

## Abstract

The robust fuzzy control for fractional-order hydroturbine regulating system is studied in this paper. First, the more practical fractional-order hydroturbine regulating system with uncertain parameters and random disturbances is presented. Then, on the basis of interval matrix theory and fractional-order stability theorem, a fuzzy control method is proposed for fractional-order hydroturbine regulating system, and the stability condition is expressed as a group of linear matrix inequalities. Furthermore, the proposed method has good robustness which can process external random disturbances and uncertain parameters. Finally, the validity and superiority are proved by the numerical simulations.


---

## Body

## 1. Introduction

Due to the advantages of describing actual projects, especially for the description of memory and hereditary properties of numerous materials and processes [1, 2], fractional calculus has attracted more and more people’s attention. It has been verified that many practical systems could be elegantly expressed with fractional calculus, such as power system [3], permanent magnet synchronous motor system [4], mechanical system [5], and chemical processing system [6].The hydroturbine regulating system is a core component for safe and stable operation of hydropower station system. For a long time up to now, the integer-order model is usually adopted [7–9]. However, as we all know, the hydroturbine regulating system is integration of hydraulic, mechanical, and electrical parts. This complex composition makes it a strong coupling, nonlinear, and nonminimum phase system [10–12]. So the integer-order model may not be suitable for perfectly describing the hydroturbine regulating system. Besides, due to the memory character and history-dependence of hydraulic servo system, the more practical fractional-order model is considered accordingly in this study. Many studies have shown that the hydroturbine regulating system may exhibit irregular nonlinear vibrations when the system is under parameter variations and external random disturbances [13–15]. Therefore, it is very important to design nonlinear controller for the stable operation of fractional-order hydroturbine regulating system. Until now, some nonlinear control schemes have been proposed for fractional-order systems, such as sliding mode control [16], predictive control [17], adaptive control [18], and pinning control [19]. However, few of the above-mentioned methods consider the uncertainty and random disturbances.Fuzzy control is a robust method, which can deal with external disturbances [20–22]. Besides, with the help of fuzzy linearization and linear matrix theory, the uncertain parameters can be well processed [23–25]. There have been many results applying fuzzy control to integer-order nonlinear systems [26–29]. However, there is little literature about fractional-order nonlinear fuzzy control and there is almost no relevant result for fractional-order hydroturbine regulating system.According to the above analysis, some advantages are shown in this study. Firstly, the fractional-order hydroturbine regulating system with uncertain parameters and random disturbances is presented, which is more in accordance with practical engineering. Secondly, a fuzzy control method is proposed for fractional-order hydroturbine regulating system, and the stability condition is expressed as a group of linear matrix inequalities (LMIs). Furthermore, the proposed method has good robustness which can process external random disturbances and uncertain parameters. Even if the system is with six uncertain parameters, the controller designed for fractional-order hydroturbine regulating system is still valid. Lastly, numerical simulations have demonstrated the effectiveness and superiority when compared with traditional PID control method.The rest of this paper is organized as follows. In Section2, the fractional-order hydroturbine regulating system is introduced. Section 3 presents the robust fuzzy controller design for fractional-order hydroturbine regulating system. Simulations are shown in Section 4. And the paper is ended in Section 5.


## 2. The Fractional-Order Hydroturbine Regulating System

The integer-order hydroturbine regulation system is given as [30](1)
d
δ
d
t
=
ω
0
ω
,
d
ω
d
t
=
1
T
a
b
m
t
-
D
ω
-
E
q
′
V
s
x
d
Σ
′
sin
⁡
δ
-
V
s
2
2
x
d
Σ
′
-
x
q
Σ
x
d
Σ
′
x
q
Σ
sin
⁡
2
δ
,
d
m
t
d
t
=
1
e
q
h
T
w
-
m
t
+
e
y
y
+
e
e
y
T
w
T
y
y
,
d
y
d
t
=
-
1
T
y
y
,where δ, ω, m
t, and y are the rotor angle deviation of the generator, the relative deviation of the rotational speed of the generator, the hydroturbine output incremental torque deviation, and the incremental deviation of the guide vane opening, respectively.The hydraulic servo system has significant historical reliance. Since it is a powerful advantage for fractional calculus to describe the function which has significant historical reliance, the fractional-order hydraulic servo system is adopted.According to fractional calculus, the fractional-order hydraulic servo system is obtained as(2)
D
α
y
=
1
T
y
u
-
y
,where T
y is the major relay connector response time.According to (1) and (2) and for convenience, we use x
1
,
x
2
,
x
3
,
x
4 to replace δ
,
ω
,
m
t
,
y. Then, the fractional-order mathematical model of hydroturbine regulation system could be represented as (3)
d
x
1
d
t
=
ω
0
x
2
,
d
x
2
d
t
=
1
T
a
b
x
3
-
D
x
2
-
E
q
′
V
s
x
d
Σ
′
sin
⁡
x
1
-
V
s
2
2
x
d
Σ
′
-
x
q
Σ
x
d
Σ
′
x
q
Σ
sin
⁡
2
x
1
,
d
x
3
d
t
=
1
e
q
h
T
w
-
x
3
+
e
y
x
4
+
e
e
y
T
w
T
y
x
4
,
d
α
x
4
d
t
α
=
-
1
T
y
x
4
,where ω
0
=
314, T
a
b
=
9.0 s, D
=
2.0, E
q
′
=
1.35, x
d
Σ
′
=
1.15, x
q
Σ
=
1.474, T
w
=
0.8 s, T
y
=
0.1 s, V
s
=
1.0, e
q
h
=
0.5, e
y
=
1.0, e
=
0.7, and α
=
0.9.Considering the universality of disturbances, the fractional-order hydroturbine regulating system could be represented as(4)
d
x
1
d
t
=
ω
0
x
2
+
rand
⁡
1
×
x
1
,
d
x
2
d
t
=
1
T
a
b
x
3
-
D
x
2
-
E
q
′
V
s
x
d
Σ
′
sin
⁡
x
1
-
V
s
2
2
x
d
Σ
′
-
x
q
Σ
x
d
Σ
′
x
q
Σ
sin
⁡
2
x
1
+
rand
⁡
1
×
x
2
,
d
x
3
d
t
=
1
e
q
h
T
w
-
x
3
+
e
y
x
4
+
e
e
y
T
w
T
y
x
4
+
rand
⁡
1
×
x
3
,
d
α
x
4
d
t
α
=
-
1
T
y
x
4
+
rand
⁡
1
×
x
4
.Figure1 shows the state trajectories of fractional-order hydroturbine regulating system (4) with initial value x
1
x
2
x
3
x
4
T
=
0.01
0.01
0.01
0.01
T. It clearly shows that the system states are in irregular and unstable nonlinear vibrations. So the effective controller should be designed for the vibration inhibition and stable operation of the fractional-order hydroturbine regulating system.Figure 1
State trajectories of fractional-order hydroturbine regulating system (4).
(a)
State trajectory ofx
1
(b)
State trajectory ofx
2
(c)
State trajectory ofx
3
(d)
State trajectory ofx
4


## 3. Robust Fuzzy Controller Design

### 3.1. Interval Matrix Theory

When the uncertain parameters are considered, the fractional-order hydroturbine regulating system could be rewritten as(5)
d
x
1
d
t
=
a
~
x
2
,
d
x
2
d
t
=
b
~
x
3
-
c
~
x
2
-
1
T
a
b
×
E
q
′
V
s
x
d
Σ
′
sin
⁡
x
1
-
1
T
a
b
×
V
s
2
2
x
d
Σ
′
-
x
q
Σ
x
d
Σ
′
x
q
Σ
sin
⁡
2
x
1
,
d
x
3
d
t
=
-
d
~
x
3
+
e
~
x
4
,
d
α
x
4
d
t
α
=
f
~
x
4
,where a
~
∈
[
a
1
,
a
2
]
,
b
~
∈
[
b
1
,
b
2
]
,
c
~
∈
[
c
1
,
c
2
]
,
d
~
∈
[
d
1
,
d
2
]
,
e
~
∈
[
e
1
,
e
2
], and f
~
∈
[
f
1
,
f
2
].To discuss the parameter uncertainties of the coefficient matrix of the fractional-order hydroturbine regulating system (5), the following interval matrix theory is introduced.The linear fractional-order interval system is given as [31] (6)
d
α
x
d
t
α
=
A
~
x
t
,where the interval uncertain matrix A
~ satisfiesA
~
∈
N
[
A
l
,
A
u
]
=
A
~
∈
R
n
×
n
∣
a
i
j
l
≤
a
i
j
≤
a
i
j
u
,
i
,
j
=
1
,
…
,
n, where A
l and A
u are the lower and upper bounds of matrix A
~, respectively.The matrixA
~ could be equivalent presented as(7)
A
~
=
A
0
i
+
E
Σ
F
,where (8)
A
0
i
=
1
2
A
i
l
+
A
i
u
,
Σ
∈
Σ
∗
=
Σ
i
∈
R
n
×
n
∣
Σ
i
=
d
i
a
g
ε
11
,
…
,
ε
1
n
,
…
,
ε
n
1
,
…
,
ε
n
n
,
ε
i
j
≤
1
,
i
,
j
=
1
,
…
,
n
,
E
=
h
11
e
1
,
…
,
h
1
n
e
1
,
…
,
h
n
1
e
n
,
…
,
h
n
n
e
n
,
F
=
h
11
e
1
,
…
,
h
1
n
e
n
,
…
,
h
n
1
e
1
,
…
,
h
n
n
e
n
T
,where H
=
(
h
i
j
)
n
×
n
=
H
i
=
1
/
2
(
A
i
u
-
A
i
l
)
,
e
i
(
i
=
1
,
…
,
n
) is the n
×
n identity matrix ith column.Note that, for anyi and Σ
∈
Σ
∗, there is(9)
Σ
Σ
T
=
Σ
T
Σ
≤
I
,
I
is
n
×
n
unit  matrix
.


### 3.2. Parallel Distributed Compensation (PDC) Controller

RuleR
i: IF z
1
(
t
) is M
i
1 and … and z
n
(
t
) is M
i
p
(10)
THEN
d
α
x
t
d
t
α
=
A
~
i
+
Δ
A
i
x
t
+
B
i
+
Δ
B
i
u
t
,
r
=
1,2
,
…
,
r
,where the fuzzy set is M
i
j
(
j
=
1,2
,
…
,
n
) and r is the IF-THEN rules number, x
(
t
)
∈
R
n is the state vector, A
i
∈
R
n
×
n, z
(
t
)
=
[
z
1
(
t
)
,
z
2
(
t
)
,
…
,
z
p
(
t
)
] are the premise variables, the fractional order is α
(
0
<
α
<
1
), and the control input is u
(
t
). The Takagi-Sugeno fuzzy model total output is introduced as(11)
d
α
x
d
t
α
=
∑
i
=
1
r
h
i
z
t
A
i
+
Δ
A
i
x
t
+
∑
i
=
1
r
h
i
z
t
B
i
+
Δ
B
i
u
t
,where(12)
h
i
z
t
=
∏
j
=
1
n
M
i
j
z
j
t
∑
i
=
1
r
∏
j
=
1
n
M
i
j
z
j
t
≥
0
,
∑
i
=
1
r
h
i
z
t
=
1
.The new fuzzy controlleru
(
t
) is designed on the basis of parallel distributed compensation (PDC) and represented as follows.RuleR
i: IF z
1
(
t
) is M
i
1 and … and z
n
(
t
) is M
i
p
(13)
THEN
u
t
=
K
i
x
t
i
=
1,2
,
…
,
rThe parallel distributed compensation controller is shown as follows:(14)
u
t
=
∑
i
=
1
r
h
i
z
t
K
i
x
t
,where K
i
(
i
=
1,2
,
…
,
r
) represents the feedback gain.By substituting (14) to (11), there follows(15)
d
α
x
d
t
α
=
∑
i
=
1
r
h
i
z
t
A
i
+
Δ
A
i
x
t
+
∑
i
=
1
r
h
i
z
t
B
i
+
Δ
B
i
∑
i
=
1
r
h
i
z
t
K
i
x
t
.According to the term∑
i
=
1
r
h
i
(
z
(
t
)
)
=
1, (15) can be equally written as(16)
d
α
x
d
t
α
=
∑
i
=
1
r
∑
j
=
1
r
h
i
z
t
h
j
z
t
A
i
+
Δ
A
i
+
B
i
+
Δ
B
i
K
j
x
t
.


### 3.3. Takagi-Sugeno Fuzzy Model of Fractional-Order Hydroturbine Regulating System

For the convenience of the coefficient matrix, the Maclaurin series expansion is introduced:(17)
sin
⁡
x
=
x
-
x
3
3
!
+
x
5
5
!
-
x
7
7
!
+
⋯
+
-
1
n
x
2
n
+
1
2
n
+
1
!
+
⋯
.Assumex
2
∈
[
-
d
,
d
]; here d
=
2. The Takagi-Sugeno fuzzy model is established, with the following two rules to describe the dynamic behavior of the system:R
1: IF x
2 is M
1
(
x
2
(
t
)
), THEN d
α
x
(
t
)
/
d
t
α
=
(
A
~
1
+
Δ
A
1
)
x
(
t
)
+
(
B
1
+
Δ
B
1
)
u
(
t
);
R
2: IF x
2 is M
2
(
x
2
(
t
)
), THEN d
α
x
(
t
)
/
d
t
α
=
(
A
~
2
+
Δ
A
2
)
x
(
t
)
+
(
B
2
+
Δ
B
2
)
u
(
t
).The membership functions are taken as follows:(18)
M
1
x
2
t
=
1
2
1
+
x
2
t
d
,
M
2
x
2
t
=
1
2
1
-
x
2
t
d
.In view of the above description, there is(19)
A
~
1
=
0
a
~
0
0
17231
16951
-
b
~
c
~
0
0
0
-
d
~
e
~
0
0
0
-
f
~
,
A
~
2
=
0
a
~
0
0
1577
16951
-
b
~
c
~
0
0
0
-
d
~
e
~
0
0
0
-
f
~
,
B
1
=
B
2
=
I
4
×
4
.So the fractional-order Takagi-Sugeno fuzzy model of the hydroturbine regulating system can be represented as(20)
d
α
x
d
t
α
=
∑
i
=
1
2
h
i
z
t
A
~
i
+
Δ
A
i
+
B
i
+
Δ
B
i
u
t
.


### 3.4. Controller Design

According to the PDC law, one has(21)
u
t
=
∑
j
=
1
2
h
j
z
t
K
j
x
t
.Then, (20) could be rewritten as (22)
d
α
x
d
t
α
=
∑
i
=
1
2
∑
j
=
1
2
h
i
x
t
h
j
x
t
A
~
i
+
Δ
A
i
+
B
i
+
Δ
B
i
K
j
x
t
.With the help of interval matrix theory in Section3.1, (22) can be equivalently given as (23)
d
α
x
d
t
α
=
∑
i
=
1
2
∑
j
=
1
2
h
i
x
t
h
j
x
t
A
0
i
+
E
Σ
F
+
Δ
A
i
+
B
i
+
Δ
B
i
K
j
x
t
.Assumption 1.
The parameter uncertainties considered here are norm-bounded in the following form:(24)
Δ
A
i
=
D
i
F
i
t
E
1
i
,
Δ
B
i
=
D
i
F
i
t
E
2
i
,where D
i
,
E
1
i
,
E
2
i are known real constant matrices of appropriate dimensions and F
i is a diagonal random matrix with Lebesgue-measurable elements and satisfies F
i
F
i
T
≤
I; I is the identity matrix with an appropriate dimension.According to Assumption1, (23) can be equally written as(25)
d
α
x
d
t
α
=
∑
i
=
1
2
∑
j
=
1
2
h
i
x
t
h
j
x
t
A
0
i
+
E
Σ
F
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
x
t
.Lemma 2 (see [32]).
For the linear fractional-order system presented below,(26)
D
α
x
=
A
x
,
x
0
=
x
0
,where A
∈
R
n
×
n
,
x
∈
R
n, and α
=
[
α
1
,
α
2
,
…
,
α
i
,
…
,
α
n
]
(
0
<
α
i
≤
1
), for all eigenvalues λ
i of matrix A, when and only when arg
⁡
(
λ
i
)
>
α
π
/
2 is satisfied; system (26) is asymptotically stable.Theorem 3.
Assume the system matrixA meets the Lyapunov function; that is, there is real positive definite symmetric matrix P as well as semipositive definite matrix Q meeting A
T
P
+
P
A
=
-
Q. The fractional-order hydroturbine regulating system (25) then will converge to the equilibrium point asymptotically.Proof.
Assumeλ is an eigenvalue of the system matrix A and the eigenvector is ξ; one can easily get(27)
A
ξ
=
λ
ξ
.
For both sides of (27), making conjugation and transpose, one has(28)
A
ξ
¯
T
=
λ
¯
ξ
T
.
For (27) left side, multiplying ξ
T
P, one can be obtain(29)
ξ
T
P
A
ξ
=
λ
ξ
T
P
ξ
.
For (28) right side, making the similar treatment and multiplying P
ξ, one can also get(30)
A
ξ
¯
T
P
ξ
=
λ
¯
ξ
T
P
ξ
.
Combining (29) and (30), one can easily get(31)
ξ
T
P
A
+
A
T
P
ξ
=
λ
+
λ
¯
ξ
T
P
ξ
.
According to Theorem3, A
T
P
+
P
A is a seminegative definite matrix. So, for any nonzero vector ξ, one has(32)
ξ
T
P
A
+
A
T
P
ξ
=
ξ
T
-
Q
ξ
≤
0
,
ξ
T
P
ξ
>
0
.
Combining (31) and (32), one can easily get(33)
λ
+
λ
¯
=
ξ
T
-
Q
ξ
ξ
T
P
ξ
¯
≤
0
.
It means, for all eigenvalues of matrixA, there is(34)
arg
⁡
λ
≥
π
2
>
α
π
2
,
α
<
1
.
Referring to Lemma2, the fractional-order hydroturbine regulating system (25) then will converge to the equilibrium point asymptotically. The proof is complete.The following more flexible theorem is presented on the basis of Theorem3.Theorem 4.
For any system variablesx
=
[
x
1
,
x
2
,
…
,
x
n
]
T, there exists a real positive definite symmetric matrix P meeting J
=
x
T
P
d
α
x
/
d
t
α
≤
0
,
(
x
T
P
d
α
x
/
d
t
α is called J function). The fractional-order hydroturbine regulating system (25) then will converge to the equilibrium point asymptotically. The condition J
=
x
T
P
d
α
x
/
d
t
α
≤
0 could be equivalently given as(35)
J
0
=
x
T
P
d
α
x
d
t
α
+
d
α
x
d
t
α
T
P
x
≤
0
.Proof.
From Theorem3, there is(36)
A
T
P
+
P
A
=
-
Q
.
According to (36), one can easily get(37)
x
T
A
T
P
+
P
A
x
=
-
x
T
Q
x
,where Q is a semipositive definite matrix. That is to say, for any system variable x, there is(38)
x
T
A
T
P
+
P
A
x
=
-
x
T
Q
x
≤
0
.
Substituting (26) to (38), one has(39)
x
T
P
d
α
x
d
t
α
+
d
α
x
d
t
α
T
P
x
≤
0
.
The positive definite symmetric matrixP is supposed as(40)
P
=
a
11
a
12
⋯
a
1
n
a
21
a
22
⋯
a
2
n
⋮
⋮
⋱
⋮
a
n
1
a
n
2
⋯
a
n
n
.
Introducing (40) to (39), one gets(41)
∑
i
,
j
=
1
n
a
i
j
x
i
d
α
x
j
d
t
α
+
∑
i
,
j
=
1
n
a
i
j
x
j
d
α
x
i
d
t
α
≤
0
.
Fora
i
j
=
a
j
i
(
∀
i
,
j
), (41) could be rewritten as(42)
∑
i
,
j
=
1
n
a
i
j
x
i
d
α
x
j
d
t
α
+
∑
i
,
j
=
1
n
a
i
j
x
j
d
α
x
i
d
t
α
=
2
∑
i
,
j
=
1
n
a
i
j
x
j
d
α
x
i
d
t
α
=
2
x
T
P
d
α
x
d
t
α
=
2
d
α
x
d
t
α
T
P
x
≤
0
.
It is clear that Theorem4 is equivalent to Theorem 3. The proof is finished.Based on the above theorems, the more practical stability conditions are proposed as follows.Theorem 5.
The fractional-order hydroturbine regulating system (25) then will converge to the equilibrium point asymptotically, once there exist a plus constant η, a positive definite matrix P, and the controller gain matrices K
i
(
i
=
1,2
) which satisfy the following LMIs: (43)
Φ
i
i
Q
F
T
F
Q
η
-
1
I
<
0
,
Φ
i
j
Q
F
T
F
Q
-
η
-
1
I
<
0
,
Q
>
0
,where(44)
Φ
i
i
=
Q
A
i
+
D
i
F
i
E
1
i
T
+
M
i
T
B
i
+
D
i
F
i
E
2
i
T
+
A
i
+
D
i
F
i
E
1
i
Q
+
B
i
+
D
i
F
i
E
2
i
M
i
+
η
E
E
T
,
Φ
i
j
=
Q
A
i
+
D
i
F
i
E
1
i
T
+
M
j
T
B
i
+
D
i
F
i
E
2
i
T
+
Q
A
j
+
D
j
F
j
E
1
j
T
+
M
i
T
B
j
+
D
j
F
j
E
2
j
T
+
A
i
+
D
i
F
i
E
1
i
Q
+
B
i
+
D
i
F
i
E
2
i
M
j
+
A
j
+
D
j
F
j
E
1
j
Q
+
B
j
+
D
j
F
j
E
2
j
M
i
+
η
E
E
T
,Q
=
P
-
1
,
M
i
=
K
i
P
-
1
,
M
j
=
K
j
P
-
1, and I is 4
×
4 unit matrix.Proof.
Based on Theorem4, choose J
0
=
x
T
P
d
α
x
/
d
t
α
+
(
d
α
x
/
d
t
α
)
T
P
x as J function for system (25).(45)
J
0
=
x
T
P
d
α
x
d
t
α
+
d
α
x
d
t
α
T
P
x
=
∑
i
=
1
4
∑
j
=
1
4
h
i
h
j
A
0
i
+
E
Σ
F
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
T
x
T
P
x
+
x
T
P
∑
i
=
1
4
∑
j
=
1
4
h
i
h
j
A
0
i
+
E
Σ
F
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
=
∑
i
=
1
4
∑
j
=
1
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
T
P
+
F
T
Σ
E
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
P
E
Σ
F
x
=
∑
i
=
1
4
h
i
2
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
i
T
P
+
F
T
Σ
E
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
i
+
P
E
Σ
F
x
+
∑
i
<
j
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
T
P
+
F
T
Σ
E
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
P
E
Σ
F
x
+
∑
i
>
j
4
h
i
h
j
x
T
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
T
P
+
F
T
Σ
E
T
P
+
P
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
+
P
E
Σ
F
x
=
∑
i
=
1
4
h
i
2
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
i
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
i
x
+
∑
i
=
1
4
h
i
2
x
T
F
T
Σ
E
T
P
+
P
E
Σ
F
x
+
2
∑
i
<
j
4
h
i
h
j
x
T
F
T
Σ
E
T
P
+
P
E
Σ
F
x
+
∑
i
<
j
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
x
+
∑
i
>
j
4
h
i
h
j
x
T
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
T
P
+
P
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
x
.
Considering that∑
i
=
1
4
h
i
2
+
2
∑
i
<
j
4
h
i
h
j
=
1 and(46)
∑
i
<
j
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
x
+
∑
i
>
j
4
h
i
h
j
x
T
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
T
P
+
P
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
x
=
∑
i
<
j
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
x
=
2
∑
i
<
j
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
2
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
2
x
,select(47)
G
i
i
=
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
1
i
K
i
,
G
i
j
=
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
1
i
K
j
+
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
2
.
There follows(48)
J
0
=
∑
i
=
1
4
h
i
2
x
T
G
i
i
T
P
+
P
G
i
i
x
+
2
∑
i
<
j
4
h
i
h
j
x
T
G
i
j
T
P
+
P
G
i
j
x
+
x
T
F
T
Σ
E
T
P
x
+
x
T
P
E
Σ
F
x
.
Selectξ
T
=
x
T
P
E
,
θ
=
F
x, and there is (49)
2
x
T
P
E
Σ
F
x
≤
η
x
T
P
E
E
T
P
x
+
η
-
1
x
T
F
T
F
x
.
Taking the transpose of both sides in (49), one obtains(50)
2
x
T
F
T
Σ
E
T
P
x
≤
η
x
T
P
E
E
T
P
x
+
η
-
1
x
T
F
T
F
x
.
According to (50) and (49), one gets (51)
x
T
F
T
Σ
E
T
P
x
+
x
T
P
E
Σ
F
x
≤
η
x
T
P
E
E
T
P
x
+
η
-
1
x
T
F
T
F
x
.
By substituting (51) into (48), one has (52)
J
0
≤
∑
i
=
1
4
h
i
2
x
T
G
i
i
T
P
+
P
G
i
i
x
+
2
∑
i
<
j
4
h
i
h
j
x
T
G
i
j
T
P
+
P
G
i
j
x
+
η
x
T
P
E
E
T
P
x
+
η
-
1
x
T
F
T
F
x
.
Note that∑
i
=
1
4
h
i
2
+
2
∑
i
<
j
4
h
i
h
j
=
1, so (52) can be equally represented as (53)
J
0
≤
∑
i
=
1
4
h
i
2
x
T
G
i
i
T
P
+
P
G
i
i
+
η
P
E
E
T
P
+
η
-
1
F
T
F
x
+
2
∑
i
<
j
4
h
i
h
j
x
T
G
i
j
T
P
+
P
G
i
j
+
η
P
E
E
T
P
+
η
-
1
F
T
F
x
.
Accordingly, (25) can be assured as long as the following inequalities hold.(54)
G
i
i
T
P
+
P
G
i
i
+
η
P
E
E
T
P
+
η
-
1
F
T
F
<
0
i
,
j
=
1,2
,
G
i
j
T
P
+
P
G
i
j
+
η
P
E
E
T
P
+
η
-
1
F
T
F
<
0
i
<
j
≤
2
.
It is clear that the fractional-order hydroturbine regulating system (25) then will converge to the equilibrium point asymptotically once (54) are satisfied.
With the help of Schur’s theorem [33], one can easily transform (54) to the standard form of linear matrix inequalities, which are presented as (43). The proof is complete.


## 3.1. Interval Matrix Theory

When the uncertain parameters are considered, the fractional-order hydroturbine regulating system could be rewritten as(5)
d
x
1
d
t
=
a
~
x
2
,
d
x
2
d
t
=
b
~
x
3
-
c
~
x
2
-
1
T
a
b
×
E
q
′
V
s
x
d
Σ
′
sin
⁡
x
1
-
1
T
a
b
×
V
s
2
2
x
d
Σ
′
-
x
q
Σ
x
d
Σ
′
x
q
Σ
sin
⁡
2
x
1
,
d
x
3
d
t
=
-
d
~
x
3
+
e
~
x
4
,
d
α
x
4
d
t
α
=
f
~
x
4
,where a
~
∈
[
a
1
,
a
2
]
,
b
~
∈
[
b
1
,
b
2
]
,
c
~
∈
[
c
1
,
c
2
]
,
d
~
∈
[
d
1
,
d
2
]
,
e
~
∈
[
e
1
,
e
2
], and f
~
∈
[
f
1
,
f
2
].To discuss the parameter uncertainties of the coefficient matrix of the fractional-order hydroturbine regulating system (5), the following interval matrix theory is introduced.The linear fractional-order interval system is given as [31] (6)
d
α
x
d
t
α
=
A
~
x
t
,where the interval uncertain matrix A
~ satisfiesA
~
∈
N
[
A
l
,
A
u
]
=
A
~
∈
R
n
×
n
∣
a
i
j
l
≤
a
i
j
≤
a
i
j
u
,
i
,
j
=
1
,
…
,
n, where A
l and A
u are the lower and upper bounds of matrix A
~, respectively.The matrixA
~ could be equivalent presented as(7)
A
~
=
A
0
i
+
E
Σ
F
,where (8)
A
0
i
=
1
2
A
i
l
+
A
i
u
,
Σ
∈
Σ
∗
=
Σ
i
∈
R
n
×
n
∣
Σ
i
=
d
i
a
g
ε
11
,
…
,
ε
1
n
,
…
,
ε
n
1
,
…
,
ε
n
n
,
ε
i
j
≤
1
,
i
,
j
=
1
,
…
,
n
,
E
=
h
11
e
1
,
…
,
h
1
n
e
1
,
…
,
h
n
1
e
n
,
…
,
h
n
n
e
n
,
F
=
h
11
e
1
,
…
,
h
1
n
e
n
,
…
,
h
n
1
e
1
,
…
,
h
n
n
e
n
T
,where H
=
(
h
i
j
)
n
×
n
=
H
i
=
1
/
2
(
A
i
u
-
A
i
l
)
,
e
i
(
i
=
1
,
…
,
n
) is the n
×
n identity matrix ith column.Note that, for anyi and Σ
∈
Σ
∗, there is(9)
Σ
Σ
T
=
Σ
T
Σ
≤
I
,
I
is
n
×
n
unit  matrix
.


## 3.2. Parallel Distributed Compensation (PDC) Controller

RuleR
i: IF z
1
(
t
) is M
i
1 and … and z
n
(
t
) is M
i
p
(10)
THEN
d
α
x
t
d
t
α
=
A
~
i
+
Δ
A
i
x
t
+
B
i
+
Δ
B
i
u
t
,
r
=
1,2
,
…
,
r
,where the fuzzy set is M
i
j
(
j
=
1,2
,
…
,
n
) and r is the IF-THEN rules number, x
(
t
)
∈
R
n is the state vector, A
i
∈
R
n
×
n, z
(
t
)
=
[
z
1
(
t
)
,
z
2
(
t
)
,
…
,
z
p
(
t
)
] are the premise variables, the fractional order is α
(
0
<
α
<
1
), and the control input is u
(
t
). The Takagi-Sugeno fuzzy model total output is introduced as(11)
d
α
x
d
t
α
=
∑
i
=
1
r
h
i
z
t
A
i
+
Δ
A
i
x
t
+
∑
i
=
1
r
h
i
z
t
B
i
+
Δ
B
i
u
t
,where(12)
h
i
z
t
=
∏
j
=
1
n
M
i
j
z
j
t
∑
i
=
1
r
∏
j
=
1
n
M
i
j
z
j
t
≥
0
,
∑
i
=
1
r
h
i
z
t
=
1
.The new fuzzy controlleru
(
t
) is designed on the basis of parallel distributed compensation (PDC) and represented as follows.RuleR
i: IF z
1
(
t
) is M
i
1 and … and z
n
(
t
) is M
i
p
(13)
THEN
u
t
=
K
i
x
t
i
=
1,2
,
…
,
rThe parallel distributed compensation controller is shown as follows:(14)
u
t
=
∑
i
=
1
r
h
i
z
t
K
i
x
t
,where K
i
(
i
=
1,2
,
…
,
r
) represents the feedback gain.By substituting (14) to (11), there follows(15)
d
α
x
d
t
α
=
∑
i
=
1
r
h
i
z
t
A
i
+
Δ
A
i
x
t
+
∑
i
=
1
r
h
i
z
t
B
i
+
Δ
B
i
∑
i
=
1
r
h
i
z
t
K
i
x
t
.According to the term∑
i
=
1
r
h
i
(
z
(
t
)
)
=
1, (15) can be equally written as(16)
d
α
x
d
t
α
=
∑
i
=
1
r
∑
j
=
1
r
h
i
z
t
h
j
z
t
A
i
+
Δ
A
i
+
B
i
+
Δ
B
i
K
j
x
t
.


## 3.3. Takagi-Sugeno Fuzzy Model of Fractional-Order Hydroturbine Regulating System

For the convenience of the coefficient matrix, the Maclaurin series expansion is introduced:(17)
sin
⁡
x
=
x
-
x
3
3
!
+
x
5
5
!
-
x
7
7
!
+
⋯
+
-
1
n
x
2
n
+
1
2
n
+
1
!
+
⋯
.Assumex
2
∈
[
-
d
,
d
]; here d
=
2. The Takagi-Sugeno fuzzy model is established, with the following two rules to describe the dynamic behavior of the system:R
1: IF x
2 is M
1
(
x
2
(
t
)
), THEN d
α
x
(
t
)
/
d
t
α
=
(
A
~
1
+
Δ
A
1
)
x
(
t
)
+
(
B
1
+
Δ
B
1
)
u
(
t
);
R
2: IF x
2 is M
2
(
x
2
(
t
)
), THEN d
α
x
(
t
)
/
d
t
α
=
(
A
~
2
+
Δ
A
2
)
x
(
t
)
+
(
B
2
+
Δ
B
2
)
u
(
t
).The membership functions are taken as follows:(18)
M
1
x
2
t
=
1
2
1
+
x
2
t
d
,
M
2
x
2
t
=
1
2
1
-
x
2
t
d
.In view of the above description, there is(19)
A
~
1
=
0
a
~
0
0
17231
16951
-
b
~
c
~
0
0
0
-
d
~
e
~
0
0
0
-
f
~
,
A
~
2
=
0
a
~
0
0
1577
16951
-
b
~
c
~
0
0
0
-
d
~
e
~
0
0
0
-
f
~
,
B
1
=
B
2
=
I
4
×
4
.So the fractional-order Takagi-Sugeno fuzzy model of the hydroturbine regulating system can be represented as(20)
d
α
x
d
t
α
=
∑
i
=
1
2
h
i
z
t
A
~
i
+
Δ
A
i
+
B
i
+
Δ
B
i
u
t
.


## 3.4. Controller Design

According to the PDC law, one has(21)
u
t
=
∑
j
=
1
2
h
j
z
t
K
j
x
t
.Then, (20) could be rewritten as (22)
d
α
x
d
t
α
=
∑
i
=
1
2
∑
j
=
1
2
h
i
x
t
h
j
x
t
A
~
i
+
Δ
A
i
+
B
i
+
Δ
B
i
K
j
x
t
.With the help of interval matrix theory in Section3.1, (22) can be equivalently given as (23)
d
α
x
d
t
α
=
∑
i
=
1
2
∑
j
=
1
2
h
i
x
t
h
j
x
t
A
0
i
+
E
Σ
F
+
Δ
A
i
+
B
i
+
Δ
B
i
K
j
x
t
.Assumption 1.
The parameter uncertainties considered here are norm-bounded in the following form:(24)
Δ
A
i
=
D
i
F
i
t
E
1
i
,
Δ
B
i
=
D
i
F
i
t
E
2
i
,where D
i
,
E
1
i
,
E
2
i are known real constant matrices of appropriate dimensions and F
i is a diagonal random matrix with Lebesgue-measurable elements and satisfies F
i
F
i
T
≤
I; I is the identity matrix with an appropriate dimension.According to Assumption1, (23) can be equally written as(25)
d
α
x
d
t
α
=
∑
i
=
1
2
∑
j
=
1
2
h
i
x
t
h
j
x
t
A
0
i
+
E
Σ
F
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
x
t
.Lemma 2 (see [32]).
For the linear fractional-order system presented below,(26)
D
α
x
=
A
x
,
x
0
=
x
0
,where A
∈
R
n
×
n
,
x
∈
R
n, and α
=
[
α
1
,
α
2
,
…
,
α
i
,
…
,
α
n
]
(
0
<
α
i
≤
1
), for all eigenvalues λ
i of matrix A, when and only when arg
⁡
(
λ
i
)
>
α
π
/
2 is satisfied; system (26) is asymptotically stable.Theorem 3.
Assume the system matrixA meets the Lyapunov function; that is, there is real positive definite symmetric matrix P as well as semipositive definite matrix Q meeting A
T
P
+
P
A
=
-
Q. The fractional-order hydroturbine regulating system (25) then will converge to the equilibrium point asymptotically.Proof.
Assumeλ is an eigenvalue of the system matrix A and the eigenvector is ξ; one can easily get(27)
A
ξ
=
λ
ξ
.
For both sides of (27), making conjugation and transpose, one has(28)
A
ξ
¯
T
=
λ
¯
ξ
T
.
For (27) left side, multiplying ξ
T
P, one can be obtain(29)
ξ
T
P
A
ξ
=
λ
ξ
T
P
ξ
.
For (28) right side, making the similar treatment and multiplying P
ξ, one can also get(30)
A
ξ
¯
T
P
ξ
=
λ
¯
ξ
T
P
ξ
.
Combining (29) and (30), one can easily get(31)
ξ
T
P
A
+
A
T
P
ξ
=
λ
+
λ
¯
ξ
T
P
ξ
.
According to Theorem3, A
T
P
+
P
A is a seminegative definite matrix. So, for any nonzero vector ξ, one has(32)
ξ
T
P
A
+
A
T
P
ξ
=
ξ
T
-
Q
ξ
≤
0
,
ξ
T
P
ξ
>
0
.
Combining (31) and (32), one can easily get(33)
λ
+
λ
¯
=
ξ
T
-
Q
ξ
ξ
T
P
ξ
¯
≤
0
.
It means, for all eigenvalues of matrixA, there is(34)
arg
⁡
λ
≥
π
2
>
α
π
2
,
α
<
1
.
Referring to Lemma2, the fractional-order hydroturbine regulating system (25) then will converge to the equilibrium point asymptotically. The proof is complete.The following more flexible theorem is presented on the basis of Theorem3.Theorem 4.
For any system variablesx
=
[
x
1
,
x
2
,
…
,
x
n
]
T, there exists a real positive definite symmetric matrix P meeting J
=
x
T
P
d
α
x
/
d
t
α
≤
0
,
(
x
T
P
d
α
x
/
d
t
α is called J function). The fractional-order hydroturbine regulating system (25) then will converge to the equilibrium point asymptotically. The condition J
=
x
T
P
d
α
x
/
d
t
α
≤
0 could be equivalently given as(35)
J
0
=
x
T
P
d
α
x
d
t
α
+
d
α
x
d
t
α
T
P
x
≤
0
.Proof.
From Theorem3, there is(36)
A
T
P
+
P
A
=
-
Q
.
According to (36), one can easily get(37)
x
T
A
T
P
+
P
A
x
=
-
x
T
Q
x
,where Q is a semipositive definite matrix. That is to say, for any system variable x, there is(38)
x
T
A
T
P
+
P
A
x
=
-
x
T
Q
x
≤
0
.
Substituting (26) to (38), one has(39)
x
T
P
d
α
x
d
t
α
+
d
α
x
d
t
α
T
P
x
≤
0
.
The positive definite symmetric matrixP is supposed as(40)
P
=
a
11
a
12
⋯
a
1
n
a
21
a
22
⋯
a
2
n
⋮
⋮
⋱
⋮
a
n
1
a
n
2
⋯
a
n
n
.
Introducing (40) to (39), one gets(41)
∑
i
,
j
=
1
n
a
i
j
x
i
d
α
x
j
d
t
α
+
∑
i
,
j
=
1
n
a
i
j
x
j
d
α
x
i
d
t
α
≤
0
.
Fora
i
j
=
a
j
i
(
∀
i
,
j
), (41) could be rewritten as(42)
∑
i
,
j
=
1
n
a
i
j
x
i
d
α
x
j
d
t
α
+
∑
i
,
j
=
1
n
a
i
j
x
j
d
α
x
i
d
t
α
=
2
∑
i
,
j
=
1
n
a
i
j
x
j
d
α
x
i
d
t
α
=
2
x
T
P
d
α
x
d
t
α
=
2
d
α
x
d
t
α
T
P
x
≤
0
.
It is clear that Theorem4 is equivalent to Theorem 3. The proof is finished.Based on the above theorems, the more practical stability conditions are proposed as follows.Theorem 5.
The fractional-order hydroturbine regulating system (25) then will converge to the equilibrium point asymptotically, once there exist a plus constant η, a positive definite matrix P, and the controller gain matrices K
i
(
i
=
1,2
) which satisfy the following LMIs: (43)
Φ
i
i
Q
F
T
F
Q
η
-
1
I
<
0
,
Φ
i
j
Q
F
T
F
Q
-
η
-
1
I
<
0
,
Q
>
0
,where(44)
Φ
i
i
=
Q
A
i
+
D
i
F
i
E
1
i
T
+
M
i
T
B
i
+
D
i
F
i
E
2
i
T
+
A
i
+
D
i
F
i
E
1
i
Q
+
B
i
+
D
i
F
i
E
2
i
M
i
+
η
E
E
T
,
Φ
i
j
=
Q
A
i
+
D
i
F
i
E
1
i
T
+
M
j
T
B
i
+
D
i
F
i
E
2
i
T
+
Q
A
j
+
D
j
F
j
E
1
j
T
+
M
i
T
B
j
+
D
j
F
j
E
2
j
T
+
A
i
+
D
i
F
i
E
1
i
Q
+
B
i
+
D
i
F
i
E
2
i
M
j
+
A
j
+
D
j
F
j
E
1
j
Q
+
B
j
+
D
j
F
j
E
2
j
M
i
+
η
E
E
T
,Q
=
P
-
1
,
M
i
=
K
i
P
-
1
,
M
j
=
K
j
P
-
1, and I is 4
×
4 unit matrix.Proof.
Based on Theorem4, choose J
0
=
x
T
P
d
α
x
/
d
t
α
+
(
d
α
x
/
d
t
α
)
T
P
x as J function for system (25).(45)
J
0
=
x
T
P
d
α
x
d
t
α
+
d
α
x
d
t
α
T
P
x
=
∑
i
=
1
4
∑
j
=
1
4
h
i
h
j
A
0
i
+
E
Σ
F
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
T
x
T
P
x
+
x
T
P
∑
i
=
1
4
∑
j
=
1
4
h
i
h
j
A
0
i
+
E
Σ
F
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
=
∑
i
=
1
4
∑
j
=
1
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
T
P
+
F
T
Σ
E
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
P
E
Σ
F
x
=
∑
i
=
1
4
h
i
2
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
i
T
P
+
F
T
Σ
E
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
i
+
P
E
Σ
F
x
+
∑
i
<
j
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
T
P
+
F
T
Σ
E
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
P
E
Σ
F
x
+
∑
i
>
j
4
h
i
h
j
x
T
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
T
P
+
F
T
Σ
E
T
P
+
P
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
+
P
E
Σ
F
x
=
∑
i
=
1
4
h
i
2
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
i
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
i
x
+
∑
i
=
1
4
h
i
2
x
T
F
T
Σ
E
T
P
+
P
E
Σ
F
x
+
2
∑
i
<
j
4
h
i
h
j
x
T
F
T
Σ
E
T
P
+
P
E
Σ
F
x
+
∑
i
<
j
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
x
+
∑
i
>
j
4
h
i
h
j
x
T
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
T
P
+
P
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
x
.
Considering that∑
i
=
1
4
h
i
2
+
2
∑
i
<
j
4
h
i
h
j
=
1 and(46)
∑
i
<
j
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
x
+
∑
i
>
j
4
h
i
h
j
x
T
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
T
P
+
P
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
x
=
∑
i
<
j
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
x
=
2
∑
i
<
j
4
h
i
h
j
x
T
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
2
T
P
+
P
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
2
i
K
j
+
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
2
x
,select(47)
G
i
i
=
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
1
i
K
i
,
G
i
j
=
A
0
i
+
D
i
F
i
E
1
i
+
B
i
+
D
i
F
i
E
1
i
K
j
+
A
0
j
+
D
j
F
j
E
1
j
+
B
j
+
D
j
F
j
E
2
j
K
i
2
.
There follows(48)
J
0
=
∑
i
=
1
4
h
i
2
x
T
G
i
i
T
P
+
P
G
i
i
x
+
2
∑
i
<
j
4
h
i
h
j
x
T
G
i
j
T
P
+
P
G
i
j
x
+
x
T
F
T
Σ
E
T
P
x
+
x
T
P
E
Σ
F
x
.
Selectξ
T
=
x
T
P
E
,
θ
=
F
x, and there is (49)
2
x
T
P
E
Σ
F
x
≤
η
x
T
P
E
E
T
P
x
+
η
-
1
x
T
F
T
F
x
.
Taking the transpose of both sides in (49), one obtains(50)
2
x
T
F
T
Σ
E
T
P
x
≤
η
x
T
P
E
E
T
P
x
+
η
-
1
x
T
F
T
F
x
.
According to (50) and (49), one gets (51)
x
T
F
T
Σ
E
T
P
x
+
x
T
P
E
Σ
F
x
≤
η
x
T
P
E
E
T
P
x
+
η
-
1
x
T
F
T
F
x
.
By substituting (51) into (48), one has (52)
J
0
≤
∑
i
=
1
4
h
i
2
x
T
G
i
i
T
P
+
P
G
i
i
x
+
2
∑
i
<
j
4
h
i
h
j
x
T
G
i
j
T
P
+
P
G
i
j
x
+
η
x
T
P
E
E
T
P
x
+
η
-
1
x
T
F
T
F
x
.
Note that∑
i
=
1
4
h
i
2
+
2
∑
i
<
j
4
h
i
h
j
=
1, so (52) can be equally represented as (53)
J
0
≤
∑
i
=
1
4
h
i
2
x
T
G
i
i
T
P
+
P
G
i
i
+
η
P
E
E
T
P
+
η
-
1
F
T
F
x
+
2
∑
i
<
j
4
h
i
h
j
x
T
G
i
j
T
P
+
P
G
i
j
+
η
P
E
E
T
P
+
η
-
1
F
T
F
x
.
Accordingly, (25) can be assured as long as the following inequalities hold.(54)
G
i
i
T
P
+
P
G
i
i
+
η
P
E
E
T
P
+
η
-
1
F
T
F
<
0
i
,
j
=
1,2
,
G
i
j
T
P
+
P
G
i
j
+
η
P
E
E
T
P
+
η
-
1
F
T
F
<
0
i
<
j
≤
2
.
It is clear that the fractional-order hydroturbine regulating system (25) then will converge to the equilibrium point asymptotically once (54) are satisfied.
With the help of Schur’s theorem [33], one can easily transform (54) to the standard form of linear matrix inequalities, which are presented as (43). The proof is complete.


## 4. Numerical Simulations

Considering the fractional-order hydroturbine regulating system (4) with uncertain parameters, a
~
=
314
+
0.1
sin
⁡
(
t
), b
~
=
2
/
9
+
0.1
cos
⁡
(
t
), c
~
=
1
/
9
+
0.1
sin
⁡
(
t
), d
~
=
5
/
2
+
0.1
cos
⁡
(
t
), e
~
=
33
/
2
+
0.1
sin
⁡
(
t
), and f
~
=
10
+
0.1
cos
⁡
(
t
).Therefore,a
~
∈
[
313.9,314.1
], b
~
∈
[
11
/
90
,
29
/
90
], c
~
∈
[
1
/
90
,
19
/
90
], d
~
∈
[
2.4,2.6
], e
~
∈
[
16.4,16.6
], and f
~
∈
[
9.9,10.1
].To take control of the fractional-order hydroturbine regulating system with uncertain parameters, we taked
=
2, A
0
i
(
i
=
1,2
), and E and F can be calculated. The value of Σ can also refer to Section 3.1.(55)
B
1
=
B
2
=
I
4
×
4
,
D
1
=
D
2
=
E
11
=
E
12
=
E
21
=
E
22
=
I
4
×
4
,
F
1
=
F
2
=
diag
⁡
diag
⁡
rand
⁡
4,4
.According to the above theorems andη
=
1000, in the environment of Matlab 7.0 LMI toolbox we acquire(56)
P
=
1
0
-
4
×
0.0000
-
0.0000
0.0000
-
0.0000
-
0.0000
0.4004
0.0013
-
0.0004
0.0000
0.0013
0.4955
0.0009
-
0.0000
0.0004
0.0093
0.4965
,
K
1
=
1
0
3
×
-
0.0007
-
7.4759
-
0.0232
0.0079
-
0.0003
-
0.0038
-
0.0000
0.0000
0.0000
0.0000
-
0.0011
-
0.0072
-
0.0000
-
0.0000
-
0.0070
0.0052
,
K
2
=
-
0.4730
-
797.0103
-
2.1527
-
0.7156
-
0.0270
-
4.2479
-
0.0467
0.0051
0.0000
0.0372
-
1.0621
-
7.2386
-
0.0000
-
0.0049
-
7.0022
5.2363
.Figure2 shows the simulation results of the traditional PID control method as well as the proposed scheme in this paper. It is clear that when the controller is applied, the system state variables quickly converge to zero point, which verifies the effectiveness. Compared with the traditional PID control method, it needs shorter transition time and the overshoot is more smaller than the designed scheme, which shows the superiority and good robustness of the proposed method.Figure 2
State trajectories of controlled fractional-order hydroturbine regulating system (4).
(a)
State trajectory ofx
1
(b)
State trajectory ofx
2
(c)
State trajectory ofx
3
(d)
State trajectory ofx
4


## 5. Conclusions

The robust fuzzy control for fractional-order hydroturbine regulating system was studied in this paper. First, mathematical model of the fractional-order hydroturbine regulating system with uncertain parameters and random disturbances was introduced. Second, a fuzzy control method was proposed for fractional-order hydroturbine regulating system. Furthermore, the stability condition of the fractional-order hydroturbine regulating system was given as a group of linear matrix inequalities and the detailed mathematical proofs were presented. Besides, the method could handle the case of time-varying parameters and random disturbances, which has shown the good robustness. Finally, the validity and superiority were verified by the simulation results.The scheme designed is simple and easy to implement and could be applied to similar fractional-order nonlinear systems. In the future, we will consider and extend the approach to other fractional-order hydroturbine governing systems, such as hydropower systems with time delay.



---

*Source: 3978050-2016-10-12.xml*