# Design of Robust Output Feedback Guaranteed Cost Control for a Class of Nonlinear Discrete-Time Systems

Authors: Yan Zhang; Yali Dong; Tianrui Li
Journal: International Journal of Engineering Mathematics
 (2014)

---

## Abstract

This paper investigates static output feedback guaranteed cost control for a class of nonlinear
discrete-time systems where the delay in state vector is inconsistent with the delay in nonlinear
perturbations. Based on the output measurement, the controller is designed to ensure the robust exponentially
stability of the closed-loop system and guarantee the performance of system to achieve an
adequate level. By using the Lyapunov-Krasovskii functional method, some sufficient conditions for
the existence of robust output feedback guaranteed cost controller are established in terms of linear
matrix inequality. A numerical example is provided to show the effectiveness of the results obtained.


---

## Body

## 1. Introduction

In control theory and practice, one of the most important open problems is the static output feedback (SOF) problem. The main principle of the SOF control is to utilize the measured output to excite the plant. Since the controller can be easily implemented in practice, the SOF control has attracted a lot of attention over the past few decades and has been applied to many areas such as economic, communication, and biological systems [1, 2]. The goal of design SOF controller is to ensure asymptotically stable or exponential stable of the original system [3]. However, in many practical systems, controller designed is to not only ensure asymptotically or exponentially stable of the system but also guarantee the performance of system to achieve an adequate level. One method of dealing with this problem is the guaranteed cost control first introduced by Chang and Peng [4]. This method has the advantage of providing an upper bound on a given performance index and thus the system performance degradation is guaranteed to be no more than this bound. Based on this idea, a lot of significant results have been addressed for continuous-time systems in [5–7] and for discrete-time systems in [8].It is well known that time-delays as well as parameter uncertainties frequently lead to instability of systems. Moreover, the existing of time-delays and uncertainties make the system more complex [9, 10].In the past studies for guaranteed cost control, almost most of the articles considered linear systems [11, 12]. However, in majority dynamic systems, the nonlinear perturbations appear more and more frequently. Therefore, we not only deal with the time-varying delays and uncertainties, but also deal with the nonlinearities. Difficulties then arise when one attempts to derive exponential stabilization conditions. Hence in this case, the methods in linear systems [11, 12] can not be directly applied to nonlinear systems. This calls for a fresh look at the problem with an improved Lyapunov-Krasovskii functionals and a new set of LMI conditions. In this paper, we aim to design robust static output feedback guaranteed cost controller for a class of nonlinear discrete-time systems with time-varying delays. By constructing a set of improved Lyapunov-Krasovskii functionals, a new criterion for the existence of robust static output feedback guaranteed cost controller is established and described in terms of linear matrix inequality. A numerical example is provided to show the effectiveness of the results obtained.Notations. In this paper, a matrix A is symmetric if A
=
A
T. λ
max
⁡
(
A
)
(
λ
min
⁡
(
A
)) denotes the maximum (minimum) value of the real parts of eigenvalues of A. The symmetric terms in a matrix are denoted by *. X
>
0 (resp., X
≥
0), for X
∈
R
n
×
n, means that the matrix is real symmetric positive definite (resp., positive semidefinite). N
+ denotes the set of all real nonnegative integers.


## 2. Preliminaries

Consider the following control system:(1)
x
(
k
+
1
)



=
[
A
+
Δ
A
(
k
)
]
x
(
k
)
+
[
A
1
+
Δ
A
1
(
k
)
]
x
(
k
-
d
(
k
)
)
+
B
u
(
k
)
+
F
f
(
x
(
k
)
)
+
G
g
(
x
(
k
-
h
(
k
)
)
)
,
y
(
k
)
=
C
x
(
k
)
+
C
1
x
(
k
-
d
(
k
)
)
,
k
∈
N
+
,
x
(
k
)
=
ϕ
k
,
k
=
-
τ
,
-
τ
+
1
,
…
,
0
,
where x
(
k
)
∈
R
n is the state vector, y
(
k
)
∈
R
r is the observation output, and u
(
k
)
∈
R
m is the control intput. A, A
1, B, F, G, C, and C
1 are given constant matrices with appropriate dimensions. Δ
A
(
k
), Δ
A
1
(
k
) are the time-varying parameter uncertainties that are assumed to satisfy the following admissible condition:
(2)
[
Δ
A
(
k
)
Δ
A
1
(
k
)
]
=
M
Δ
(
k
)
[
N
1
N
2
]
,
Δ
T
(
k
)
Δ
(
k
)
⩽
I
,
∀
k
∈
N
+
,
where M, N
1, and N
2 are some given constant matrices with appropriate dimensions. The positive integers d
(
k
) and h
(
k
) are time-varying delays satisfying
(3)
0
≤
d
(
k
)
≤
d
,
0
≤
h
(
k
)
≤
h
,
where d, h are known positive integers. f
(
x
(
k
)
)
∈
R
n and g
(
x
(
k
-
h
(
k
)
)
∈
R
n are unknown nonlinear functions, assumed as
(4)
f
T
(
x
(
k
)
)
f
(
x
(
k
)
)
⩽
β
1
x
T
(
k
)
L
1
T
L
1
x
(
k
)
,
g
T
(
x
(
k
-
h
(
k
)
)
)
g
(
x
(
k
-
h
(
k
)
)
)
⩽
β
2
x
T
(
k
-
h
(
k
)
)
L
2
T
L
2
x
(
k
-
h
(
k
)
)
,
where β
1, β
2 are known positive integers and L
1, L
2 are known real matrices. The initial condition ϕ
=
(
ϕ
-
τ
,
ϕ
-
τ
+
1
,
…
,
ϕ
0
)
∈
R
(
τ
+
1
)
n with the norm
(5)
∥
ϕ
∥
=
max
⁡
{
∥
x
(
-
τ
)
∥
,
…
,
∥
x
(
0
)
∥
}
,
where τ
=
max
⁡
{
h
,
d
}.The corresponding cost function is as follows:(6)
J
=
∑
k
=
0
∞
‍
{
x
T
(
k
)
S
x
(
k
)
+
x
T
(
k
-
d
(
k
)
)
S
1
x
(
k
-
d
(
k
)
)
+
x
T
(
k
-
h
(
k
)
)
S
2
x
(
k
-
h
(
k
)
)
+
u
T
(
k
)
R
u
(
k
)
}
,
where S, S
1, S
2, and R are given symmetric positive definite matrices with appropriate dimensions.Substituting the output feedback controlleru
(
k
)
=
K
y
(
k
) into system (1), we have
(7)
x
(
k
+
1
)
=
[
A
~
(
k
)
+
B
K
C
]
x
(
k
)
+
[
A
~
1
(
k
)
+
B
K
C
1
]
x
(
k
-
d
(
k
)
)
+
F
f
(
x
(
k
)
)
+
G
g
(
x
(
k
-
h
(
k
)
)
)
,
where A
~
(
k
)
=
A
+
Δ
A
(
k
) and A
~
1
(
k
)
=
A
1
+
Δ
A
1
(
k
).The objective of this paper is to design an output feedback controlleru
(
k
)
=
K
y
(
k
) for system (1) and cost function (6) such that the resulting closed-loop system is robust exponentially stable with an upper bound for cost function (6).We first give the following definitions, which will be used in the next theorems and proofs.Definition 1.
Givenα
>
0, the closed-loop system (7) is said to be robust exponentially stable with a decay rate α, if there exists scalars σ
>
0 such that for every solution x
(
k
,
ϕ
) of the system satisfies the condition:
(8)
∥
x
(
k
,
ϕ
)
∥
≤
σ
e
-
α
k
∥
ϕ
∥
,
∀
k
∈
N
+
.Definition 2.
For system (1) and cost function (6), if there exist a static output feedback control law u
*
(
k
) and a positive constant J
* such that the closed-loop system (7) is robust exponentially stable with a decay rate α and the value (6) satisfies J
≤
J
*, then J
* is said to be a guaranteed cost index and u
*
(
k
) is said to be a robust output feedback guaranteed cost control law of the system.The following lemmas are essential in establishing our main results.Lemma 3 (see [11]).
For anyx, y
∈
R
n, and positive symmetric definite matrix N
∈
R
n
×
n, we have
(9)
±
2
y
T
x
≤
x
T
N
-
1
x
+
y
T
N
y
.Lemma 4 (Schur complement lemma [13]).
Given constant matrixX, Y, and Z with appropriate dimensions satisfying X
=
X
T, Y
=
Y
T
>
0. Then X
+
Z
T
Y
-
1
Z
<
0 if and only if S
=
(
X
Z
T
Z
-
Y
)
<
0.


## 3. Main Results

In this section, by constructing a new set of Lyapunov-Krasovskii functionals, we give a sufficient condition for the existence of robustly output feedback guaranteed cost control for system (1).Theorem 5.
For a given scalarα
>
0, the control u
(
k
)
=
K
y
(
k
) is a robustly static output feedback guaranteed cost controller for nonlinear system (1), if there exist symmetric positive definite matrices P, Q
1, Q
2, R
1, R
2, and K, arbitrary matrix N, and scalars ε
1
≥
0, ε
2
≥
0, such that the following LMI holds:
(10)
(
Θ
1
0
Θ
13
Θ
14
Θ
15
*
Θ
2
0
0
0
*
*
Θ
3
0
0
*
*
*
Θ
4
0
*
*
*
*
Θ
5
)
<
0
,
where
(11)
Θ
1
=
(
Δ
11
Δ
12
0
*
Δ
22
Δ
23
*
*
Δ
33
)
,
Θ
14
=
(
0
0
0
0
NB
NM
NF
NG
0
0
0
0
)
,
Θ
13
=
(
C
T
K
T
C
T
K
T
0
0
0
0
)
,
Θ
15
=
(
0
0
0
0
C
1
T
K
T
C
1
T
K
T
)
,
Θ
2
=
diag
⁡
{
Δ
44
,
Δ
55
,
Δ
66
}
,
Θ
3
=
diag
⁡
{
-
0.5
R
-
1
,
-
I
}
,
Θ
4
=
diag
⁡
{
Λ
1
,
Λ
2
,
-
I
,
-
I
}
,
Θ
5
=
diag
⁡
{
-
0.5
e
-
2
α
d
R
-
1
,
-
e
-
2
α
I
}
,
Δ
11
=
-
P
+
d
¯
(
Q
1
+
R
1
)
+
e
α
N
1
T
N
1
+
h
¯
(
Q
2
+
R
2
)
+
β
1
(
e
2
α
+
ε
1
)
L
1
T
L
1
+
S
,
Δ
12
=
e
α
N
A
,
Δ
23
=
e
α
d
¯
N
A
1
,
Δ
33
=
-
Q
1
-
R
1
+
e
α
d
¯
N
2
T
N
2
+
e
2
α
d
S
1
,
Δ
22
=
P
-
(
N
+
N
T
)
,
Δ
44
=
-
Q
2
-
R
2
+
(
ε
2
+
1
)
β
2
e
2
α
h
L
2
T
L
2
+
e
2
α
h
S
2
,
Δ
55
=
-
ε
1
I
,
Δ
66
=
-
ε
2
I
,
Λ
1
=
-
e
-
2
α
d
¯
I
,
Λ
2
=
-
1
e
α
d
¯
+
e
α
I
,
d
¯
=
d
+
1
,
h
¯
=
h
+
1
,
and the guaranteed cost value is given by J
*
=
μ
2
∥
ϕ
∥
2, where
(12)
μ
2
=
λ
max
⁡
(
P
)
+
d
(
d
+
1
)
[
λ
max
⁡
(
Q
1
)
+
λ
max
⁡
(
R
1
)
]
+
h
(
h
+
1
)
[
λ
max
⁡
(
Q
2
)
+
λ
max
⁡
(
R
2
)
]
.Proof.
We first introduce the new variablez
(
k
)
=
e
α
k
x
(
k
). The closed-loop system (7) is reduced to
(13)
z
(
k
+
1
)
=
[
A
~
(
k
)
+
B
K
C
]
e
α
z
(
k
)
+
[
A
~
1
(
k
)
+
B
K
C
1
]
e
α
(
d
(
k
)
+
1
)
z
(
k
-
d
(
k
)
)
+
F
e
α
(
k
+
1
)
f
¯
(
z
(
k
)
)
+
G
e
α
(
k
+
1
)
g
¯
(
z
(
k
-
h
(
k
)
)
)
,
where f
¯
(
z
(
k
)
)
=
f
(
z
(
k
)
/
e
α
k
) and g
¯
(
z
(
k
-
h
(
k
)
)
)
=
g
(
z
(
k
-
h
(
k
)
)
/
e
α
(
k
-
h
(
k
)
)
).
Associated with (2), the above equality is reduced to
(14)
z
(
k
+
1
)
=
[
A
¯
(
k
)
+
B
¯
(
k
)
K
C
¯
(
k
)
]
z
(
k
)
+
[
A
1
¯
(
k
)
+
B
¯
(
k
)
K
C
1
¯
(
k
)
]
z
(
k
-
d
(
k
)
)
+
F
¯
(
k
)
f
¯
(
z
(
k
)
)
+
G
¯
(
k
)
g
¯
(
z
(
k
-
h
(
k
)
)
)
,
where
(15)
A
¯
(
k
)
=
e
α
[
A
+
M
Δ
(
k
)
N
1
]
,
C
¯
(
k
)
=
e
-
α
k
C
,
A
1
¯
(
k
)
=
e
α
(
d
(
k
)
+
1
)
[
A
1
+
M
Δ
(
k
)
N
2
]
,
B
¯
(
k
)
=
e
α
(
k
+
1
)
B
,
C
1
¯
(
k
)
=
e
-
α
(
k
-
d
(
k
)
)
C
1
,
F
¯
(
k
)
=
e
α
(
k
+
1
)
F
,
G
¯
(
k
)
=
e
α
(
k
+
1
)
G
,
(16)
f
¯
T
(
z
(
k
)
)
f
¯
(
z
(
k
)
)
≤
β
1
e
-
2
α
k
z
T
(
k
)
L
1
T
L
1
z
(
k
)
,
(17)
g
¯
T
(
z
(
k
-
h
(
k
)
)
)
g
¯
(
z
(
k
-
h
(
k
)
)
)
≤
β
2
e
-
2
α
(
k
-
h
(
k
)
)
z
T
(
k
-
h
(
k
)
)
L
2
T
L
2
z
(
k
-
h
(
k
)
)
.
Consider a Lyapunov-Krasovskii functional candidate for the closed-loop system (14) as
(18)
V
(
k
)
=
V
1
(
k
)
+
V
2
(
k
)
+
V
3
(
k
)
+
V
4
(
k
)
+
V
5
(
k
)
,
where
(19)
V
1
(
k
)
=
z
T
(
k
)
P
z
(
k
)
,
V
2
(
k
)
=
∑
i
=
k
-
d
(
k
)
k
-
1
‍
z
T
(
i
)
Q
1
z
(
i
)
+
∑
i
=
k
-
h
(
k
)
k
-
1
‍
z
T
(
i
)
Q
2
z
(
i
)
,
V
3
(
k
)
=
∑
j
=
-
d
+
1
0
‍
∑
i
=
k
+
j
k
-
1
‍
z
T
(
i
)
Q
1
z
(
i
)
+
∑
j
=
-
h
+
1
0
‍
∑
i
=
k
+
j
k
-
1
‍
z
T
(
i
)
Q
2
z
(
i
)
,
V
4
(
k
)
=
∑
i
=
k
-
d
(
k
)
k
-
1
‍
z
T
(
i
)
R
1
z
(
i
)
+
∑
i
=
k
-
h
(
k
)
k
-
1
‍
z
T
(
i
)
R
2
z
(
i
)
,
V
5
(
k
)
=
∑
j
=
-
d
+
1
0
‍
∑
i
=
k
+
j
k
-
1
‍
z
T
(
i
)
R
1
z
(
i
)
+
∑
j
=
-
h
+
1
0
‍
∑
i
=
k
+
j
k
-
1
‍
z
T
(
i
)
R
2
z
(
i
)
.
Calculating the difference of V
(
k
) we have
(20)
Δ
V
1
(
k
)
=
z
T
(
k
+
1
)
P
z
(
k
+
1
)
-
z
T
(
k
)
P
z
(
k
)
,
(21)
Δ
V
2
(
k
)
≤
z
T
(
k
)
(
Q
1
+
Q
2
)
z
(
k
)
-
z
T
(
k
-
d
(
k
)
)
Q
1
z
(
k
-
d
(
k
)
)
-
z
T
(
k
-
h
(
k
)
)
Q
2
z
(
k
-
h
(
k
)
)
+
∑
i
=
k
-
d
+
1
k
‍
z
T
(
i
)
Q
1
z
(
i
)
+
∑
i
=
k
-
h
+
1
k
‍
z
T
(
i
)
Q
2
z
(
i
)
,
(22)
Δ
V
3
(
k
)
=
∑
j
=
-
d
+
1
0
‍
{
∑
i
=
k
+
j
+
1
k
‍
z
T
(
i
)
Q
1
z
(
i
)
-
∑
i
=
k
+
j
k
-
1
‍
z
T
(
i
)
Q
1
z
(
i
)
}
+
∑
j
=
-
h
+
1
0
‍
{
∑
i
=
k
+
j
+
1
k
‍
z
T
(
i
)
Q
2
z
(
i
)
-
∑
i
=
k
+
j
k
-
1
‍
z
T
(
i
)
Q
2
z
(
i
)
}
=
z
T
(
k
)
(
d
Q
1
+
h
Q
2
)
z
(
k
)
-
∑
j
=
k
-
d
+
1
k
‍
z
T
(
j
)
Q
1
z
(
j
)
-
∑
j
=
k
-
h
+
1
k
‍
z
T
(
j
)
Q
2
z
(
j
)
.
Combine (21) and (22), we have
(23)
Δ
V
2
(
k
)
+
Δ
V
3
(
k
)
≤
z
T
(
k
)
(
d
¯
Q
1
+
h
¯
Q
2
)
z
(
k
)
-
z
T
(
k
-
d
(
k
)
)
Q
1
z
(
k
-
d
(
k
)
)
-
z
T
(
k
-
h
(
k
)
)
Q
2
z
(
k
-
h
(
k
)
)
.
Similarly, we can get
(24)
Δ
V
4
(
k
)
+
Δ
V
5
(
k
)
≤
z
T
(
k
)
(
d
¯
R
1
+
h
¯
R
2
)
z
(
k
)
-
z
T
(
k
-
d
(
k
)
)
R
1
z
(
k
-
d
(
k
)
)
-
z
T
(
k
-
h
(
k
)
)
R
2
z
(
k
-
h
(
k
)
)
.
Therefore, from (17)–(24), we have
(25)
Δ
V
(
k
)
≤
z
T
(
k
+
1
)
P
z
(
k
+
1
)
+
z
T
(
k
)
[
d
¯
(
Q
1
+
R
1
)
+
h
¯
(
Q
2
+
R
2
)
-
P
]
z
(
k
)
-
z
T
(
k
-
d
(
k
)
)
(
Q
1
+
R
1
)
z
(
k
-
d
(
k
)
)
-
z
T
(
k
-
h
(
k
)
)
(
Q
2
+
R
2
)
z
(
k
-
h
(
k
)
)
.
Multiplying 2
z
T
(
k
+
1
)
N both sides of the identity (14),
(26)
-
2
z
T
(
k
+
1
)
N
z
(
k
+
1
)
+
2
z
T
(
k
+
1
)
×
N
(
A
¯
(
k
)
+
B
¯
(
k
)
K
C
¯
(
k
)
)
z
(
k
)
+
2
z
T
(
k
+
1
)
×
N
(
A
1
¯
(
k
)
+
B
¯
(
k
)
K
C
1
¯
(
k
)
)
z
(
k
-
d
(
k
)
)
+
2
z
T
(
k
+
1
)
N
F
¯
(
k
)
f
¯
+
2
z
T
(
k
+
1
)
N
G
¯
(
k
)
g
¯
=
0
.
Note that for any ε
1
≥
0,  ε
2
≥
0, it follows from (16) to (17)
(27)
ε
1
[
β
1
e
-
2
α
k
z
T
(
k
)
L
1
T
L
1
z
(
k
)
-
f
¯
T
(
z
(
k
)
)
f
¯
(
z
(
k
)
)
]
≥
0
,
ε
2
[
(
z
(
k
-
h
(
k
)
)
)
β
2
e
-
2
α
(
k
-
h
(
k
)
)
z
T
(
k
-
h
(
k
)
)
L
2
T
L
2
z
(
k
-
h
(
k
)
)
-
g
¯
T
(
z
(
k
-
h
(
k
)
)
)
g
¯
(
z
(
k
-
h
(
k
)
)
)
]
≥
0
.
Substituting (26)-(27) into (25), we have
(28)
Δ
V
(
k
)
≤
z
T
(
k
+
1
)
P
z
(
k
+
1
)
+
z
T
(
k
)
[
d
¯
(
Q
1
+
R
1
)
+
h
¯
(
Q
2
+
R
2
)
-
P
]
z
(
k
)
-
z
T
(
k
-
d
(
k
)
)
(
Q
1
+
R
1
)
z
(
k
-
d
(
k
)
)
-
z
T
(
k
-
h
(
k
)
)
(
Q
2
+
R
2
)
z
(
k
-
h
(
k
)
)
-
2
z
T
(
k
+
1
)
N
z
(
k
+
1
)
+
2
z
T
(
k
+
1
)
×
N
(
A
¯
(
k
)
+
B
¯
(
k
)
K
C
¯
(
k
)
)
z
(
k
)
+
2
z
T
(
k
+
1
)
×
N
(
A
1
¯
(
k
)
+
B
¯
(
k
)
K
C
1
¯
(
k
)
)
z
(
k
-
d
(
k
)
)
+
2
z
T
(
k
+
1
)
N
F
¯
(
k
)
f
¯
(
z
(
k
)
)
+
2
z
T
(
k
+
1
)
N
G
¯
(
k
)
g
¯
(
z
(
k
-
h
(
k
)
)
)
+
ε
1
[
β
1
e
-
2
α
k
z
T
(
k
)
L
1
T
L
1
z
(
k
)

-

f
¯
T
(
z
(
k
)
)
f
¯
(
z
(
k
)
)
]
+
ε
2
[
(
z
(
k
-
h
(
k
)
)
)
β
2
e
-
2
α
(
k
-
h
(
k
)
)
z
T
(
k
-
h
(
k
)
)
L
2
T
L
2
z
(
k
-
h
(
k
)
)
-
g
¯
T
(
z
(
k
-
h
(
k
)
)
)
g
¯
(
z
(
k
-
h
(
k
)
)
)
]
.
Dealing with partial idem in (28) using Lemma 3, it follows
(29)
2
z
T
(
k
+
1
)
N
(
A
¯
(
k
)
+
B
¯
(
k
)
K
C
¯
(
k
)
)
z
(
k
)
=
2
z
T
(
k
+
1
)
e
α
[
N
A
+
N
M
Δ
(
k
)
N
1
]
z
(
k
)
+
2
z
T
(
k
+
1
)
e
α
N
B
K
C
z
(
k
)
≤
2
z
T
(
k
+
1
)
e
α
N
A
z
(
k
)
+
e
α
z
T
(
k
+
1
)
N
M
M
T
N
T
z
(
k
+
1
)
+
e
α
z
T
(
k
)
N
1
T
N
1
z
(
k
)
+
z
T
(
k
+
1
)
e
2
α
N
B
B
T
N
T
z
(
k
+
1
)
+
z
T
(
k
)
C
T
K
T
K
C
z
(
k
)
,
2
z
T
(
k
+
1
)
N
(
A
1
¯
(
k
)
+
B
¯
(
k
)
K
C
1
¯
(
k
)
)
z
(
k
-
d
(
k
)
)
=
z
T
(
k
+
1
)
e
α
(
d
(
k
)
+
1
)
(
N
A
1
+
A
1
T
N
T
)
z
(
k
-
d
(
k
)
)
+
2
e
α
(
d
(
k
)
+
1
)
z
T
(
k
+
1
)
N
M
Δ
(
k
)
N
2
z
(
k
-
d
(
k
)
)
+
2
z
T
(
k
+
1
)
e
α
d
(
k
)
N
B
e
α
K
C
1
z
(
k
-
d
(
k
)
)
≤
z
T
(
k
+
1
)
e
α
d
¯
(
N
A
1
+
A
1
T
N
T
)
z
(
k
-
d
(
k
)
)
+
e
α
d
¯
z
T
(
k
+
1
)
N
M
M
T
N
T
z
(
k
+
1
)
+
e
α
d
¯
z
T
(
k
-
d
(
k
)
)
N
2
T
N
2
z
(
k
-
d
(
k
)
)
+
z
T
(
k
+
1
)
e
2
α
d
N
B
B
T
N
T
z
(
k
+
1
)
+
e
2
α
z
T
(
k
-
d
(
k
)
)
C
1
T
K
T
K
C
1
z
(
k
-
d
(
k
)
)
,
2
z
T
(
k
+
1
)
N
F
¯
(
k
)
f
¯
(
z
(
k
)
)
=
2
z
T
(
k
+
1
)
N
F
e
α
(
k
+
1
)
f
¯
(
z
(
k
)
)
≤
z
T
(
k
+
1
)
N
F
F
T
N
T
z
(
k
+
1
)
+
e
2
α
(
k
+
1
)
f
¯
T
(
z
(
k
)
)
f
¯
(
z
(
k
)
)
≤
z
T
(
k
+
1
)
N
F
F
T
N
T
z
(
k
+
1
)
+
β
1
e
2
α
z
T
(
k
)
L
1
T
L
1
z
(
k
)
.
Similarly, we have
(30)
2
z
T
(
k
+
1
)
N
G
¯
(
k
)
g
¯
(
z
(
k
-
h
(
k
)
)
)
≤
z
T
(
k
+
1
)
N
G
G
T
N
T
z
(
k
+
1
)
+
β
2
e
2
α
h
z
T
(
k
-
h
(
k
)
)
L
2
T
L
2
z
(
k
-
h
(
k
)
)
.
Adding the following relation to inequality (28)
(31)
[
z
T
(
k
)
S
¯
z
(
k
)
+
z
T
(
k
-
d
(
k
)
)
S
1
¯
z
(
k
-
d
(
k
)
)
+
z
T
(
k
-
h
(
k
)
)
S
2
¯
z
(
k
-
h
(
k
)
)
+
u
T
(
k
)
R
u
(
k
)
]
-
[
z
T
(
k
)
S
¯
z
(
k
)
+
z
T
(
k
-
d
(
k
)
)
S
1
¯
z
(
k
-
d
(
k
)
)
+
z
T
(
k
-
h
(
k
)
)
S
2
¯
z
(
k
-
h
(
k
)
)
+
u
T
(
k
)
R
u
(
k
)
]
=
0
,
where S
¯
=
e
-
2
α
k
S, S
1
¯
=
e
-
2
α
(
k
-
d
(
k
)
)
S
1, and S
2
¯
=
e
-
2
α
(
k
-
h
(
k
)
)
S
2, and using
(32)
u
T
(
k
)
R
u
(
k
)
≤
z
T
(
k
)
C
T
K
T
R
K
C
z
(
k
)
+
2
e
α
d
z
T
(
k
)
C
T
K
T
R
K
C
1
z
(
k
-
d
(
k
)
)
+
e
2
α
d
z
T
(
k
-
d
(
k
)
)
C
1
T
K
T
R
K
C
1
z
(
k
-
d
(
k
)
)
≤
2
z
T
(
k
)
C
T
K
T
R
K
C
z
(
k
)
+
2
e
2
α
d
z
T
(
k
-
d
(
k
)
)
C
1
T
K
T
R
K
C
1
z
(
k
-
d
(
k
)
)
,
we can get
(33)
Δ
V
(
k
)
≤
ξ
T
(
k
)
Ω
ξ
(
k
)
-
[
z
T
(
k
)
S
¯
z
(
k
)
+
z
T
(
k
-
d
(
k
)
)
S
1
¯
z
(
k
-
d
(
k
)
)
+
z
T
(
k
-
h
(
k
)
)
S
2
¯
z
(
k
-
h
(
k
)
)
+
u
T
(
k
)
R
u
(
k
)
]
,
where ξ
(
k
)
=
[
z
T
(
k
) z
T
(
k
+
1
) z
T
(
k
-
d
(
k
)
) z
T
(
k
-
h
(
k
)
) f
¯
T
(
z
(
k
)
) g
¯
T
(
z
(
k
-
h
(
k
)
)
)
]
T and
(34)
Ω
=
(
Ω
11
Δ
12
0
0
0
0
*
Ω
22
Δ
23
0
0
0
*
*
Ω
33
0
0
0
*
*
*
Δ
44
0
0
*
*
*
*
Δ
55
0
*
*
*
*
*
Δ
66
)
,
Ω
11
=
Δ
11
+
C
T
K
T
(
I
+
R
+
R
T
)
K
C
,
Ω
22
=
Δ
22
+
e
2
α
d
¯
N
B
B
T
N
T
+
e
α
N
M
M
T
N
T
+
e
α
d
¯
N
M
M
T
N
T
+
N
F
F
T
N
T
+
N
G
G
T
N
T
,
Ω
33
=
Δ
33
+
e
2
α
C
1
T
K
T
K
C
1
+
2
e
2
α
d
C
1
T
K
T
R
K
C
1
.
By Lemma4, the condition Ω
<
0 is equivalent to LMI (10). Therefore, from (33) it follows that
(35)
Δ
V
(
k
)
<
0
,
which implies that V
(
k
)
≤
V
(
0
), ∀
k
∈
N
+.
We can easily get(36)
μ
1
∥
z
(
k
)
∥
2
≤
V
(
k
)
≤
μ
2
∥
z
k
∥
2
,
where ∥
z
k
∥
=
max
⁡
{
∥
z
(
k
-
τ
)
∥
,
…
,
∥
z
(
k
)
∥
},  μ
1
=
λ
min
⁡
(
P
),
(37)
μ
2
=
λ
max
⁡
(
P
)
+
d
(
d
+
1
)
[
λ
max
⁡
(
Q
1
)
+
λ
max
⁡
(
R
1
)
]
+
h
(
h
+
1
)
[
λ
max
⁡
(
Q
2
)
+
λ
max
⁡
(
R
2
)
]
.
From (35) and (36), we can get
(38)
μ
1
∥
z
(
k
)
∥
2
≤
μ
2
∥
ϕ
¯
∥
2
,
∥
z
(
k
)
∥
≤
μ
2
μ
1
∥
ϕ
¯
∥
.
Using the relation z
(
k
)
=
e
α
k
x
(
k
), we can get
(39)
∥
x
(
k
)
∥
≤
μ
2
μ
1
e
-
α
k
∥
ϕ
∥
,
∀
k
∈
N
+
.
Therefore, the closed-loop system (7) is exponentially stable. Next we will find the guaranteed cost value, from (33), we can get
(40)
Δ
V
(
k
)
≤
-
[
x
T
(
k
)
S
x
(
k
)
+
x
T
(
k
-
d
(
k
)
)
S
1
x
(
k
-
d
(
k
)
)
+
x
T
(
k
-
h
(
k
)
)
S
2
x
(
k
-
h
(
k
)
)
+
u
T
(
k
)
R
u
(
k
)
]
.
Summing up both sides of (40) from 0 to n
-
1, we can get
(41)
∑
k
=
0
n
-
1
‍
[
x
T
(
k
)
S
x
(
k
)
+
x
T
(
k
-
d
(
k
)
)
S
1
x
(
k
-
d
(
k
)
)
+
x
T
(
k
-
h
(
k
)
)
S
2
x
(
k
-
h
(
k
)
)
+
u
T
(
k
)
R
u
(
k
)
]
≤
V
(
0
)
-
V
(
n
)
.
Letting n
→
+
∞, noting that V
(
n
)
→
0, we can get
(42)
J
=
∑
k
=
0
∞
‍
[
x
T
(
k
)
S
x
(
k
)
+
x
T
(
k
-
d
(
k
)
)
S
1
x
(
k
-
d
(
k
)
)
+
x
T
(
k
-
h
(
k
)
)
S
2
x
(
k
-
h
(
k
)
)
+
u
T
(
k
)
R
u
(
k
)
x
T
(
k
)
]
≤
V
(
0
)
,
associated with (36), and we have J
≤
μ
2
∥
ϕ
∥
2
=
J
*.Remark 6.
When time-delay in state vector keeps consistent with the delay in nonlinear perturbations and uncertain items disappear, the system (1) induced to
(43)
x
(
k
+
1
)
=
A
x
(
k
)
+
A
1
x
(
k
-
d
(
k
)
)
+
B
u
(
k
)
+
F
f
(
x
(
k
)
)
+
G
g
(
x
(
k
-
d
(
k
)
)
)
,
y
(
k
)
=
C
x
(
k
)
+
C
1
x
(
k
-
d
(
k
)
)
,
k
∈
N
+
,
x
(
k
)
=
ϕ
k
,
k
=
-
d
,
-
d
+
1
,
…
,
0
,
at the same time, the closed-loop system (7), and cost function (6) are reduced to
(44)
x
(
k
+
1
)
=
[
A
+
B
K
C
]
x
(
k
)
+
[
A
1
+
B
K
C
1
]
x
(
k
-
d
(
k
)
)
+
F
f
(
x
(
k
)
)
+
G
g
(
x
(
k
-
d
(
k
)
)
)
,
J
=
∑
k
=
0
∞
‍
{
x
T
(
k
)
S
x
(
k
)
+
x
T
(
k
-
d
(
k
)
)
S
1
+
u
T
(
k
)
R
u
(
k
)
}
.
Then we give a sufficient condition for the existence of static output feedback control for system (43).Theorem 7.
For a given scalarα
>
0, the control u
(
k
)
=
K
y
(
k
) is a static output feedback guaranteed cost controller for system (43), if there exist symmetric positive definite matrices P, Q
1, R
1, and K, arbitrary matrix N, and scalars ε
1
≥
0, ε
2
≥
0, such that the following LMI holds:
(45)
(
Γ
1
0
Θ
13
Γ
14
Γ
15
*
Γ
2
0
0
0
*
*
Θ
3
0
0
*
*
*
Γ
4
0
*
*
*
*
Γ
5
)
<
0
,
where
(46)
Γ
1
=
(
Δ
11
′
Δ
12
0
*
Δ
22
Δ
23
*
*
Δ
33
′
)
,
Γ
14
=
(
0
0
0
N
B
N
F
N
G
0
0
0
)
,
Θ
15
=
(
0
0
0
0
C
1
T
K
T
C
1
T
K
T
)
,
Γ
2
=
diag
⁡
{
Δ
44
′
,
Δ
55
′
}
,
Γ
4
=
diag
⁡
{
Λ
1
,
Λ
2
,
-
I
,
-
I
}
,
Γ
5
=
diag
⁡
{
-
0.5
e
-
2
α
d
R
-
1
,
-
e
-
2
α
I
}
,
Δ
11
′
=
-
P
+
d
¯
(
Q
1
+
R
1
)
+
β
1
(
e
2
α
+
ε
1
)
L
1
T
L
1
+
S
,
Δ
33
′
=
-
Q
1
-
R
1
+
e
2
α
d
S
1
+
β
2
(
ε
2
+
e
2
α
d
)
L
2
T
L
2
,
Δ
44
′
=
-
ε
1
I
,
Δ
55
′
=
-
ε
2
I
,
and the guaranteed cost value is given by J
*
=
μ
2
∥
ϕ
∥
2, where μ
2
′
=
λ
max
⁡
(
P
)
+
d
(
d
+
1
)
[
λ
max
⁡
(
Q
1
)
+
λ
max
⁡
(
R
1
)
].Proof.
The proof of Theorem7 is similar to that of Theorem 5, which are omitted.Remark 8.
In this paper, we design the controller directly from the LMI without variable transformation [11] which reduces the amount of calculation. Moreover, based on Theorem 5, one can deduce the criteria for linear discrete-time systems with time-delay and nonlinear discrete-time systems with constant time-delay.


## 4. Numerical Example

Consider the nonlinear uncertain discrete-time system (1) with the following parameters:
(47)
A
=
(
-
0.02
0.12
-
0.1
0.1
0.03
0.1
-
0.1
0.03
0.2
)
,
A
1
=
(
0.02
0.03
0.02
0.01
0.02
-
0.01
-
0.05
0.02
0.08
)
,
B
=
(
-
0.05
0.1
0.2
0.1
0.32
0.1
0.05
-
0.06
0.12
)
,
F
=
(
0.16
-
0.3
0.02
-
0.2
0.14
0.2
0.13
0.14
0.1
)
,
G
=
(
-
0.03
0.06
0.02
0.4
0.1
0.1
0.02
-
0.03
0.02
)
,
C
=
(
0.12
0.28
0
-
0.06
0.09
0.02
0.2
0.03
0.2
)
,
C
1
=
(
0.04
-
0.06
0.07
0.05
0.08
0.07
0.04
0.01
0.05
)
,
N
1
=
(
-
0.5
0.2
0.1
0.2
0.3
0.1
-
0.8
0.2
0.1
)
,
N
2
=
(
-
0.6
0.3
0.1
-
0.1
-
0.2
0.2
-
0.1
-
0.5
0.3
)
,
M
=
(
0.03
0.02
0.04
-
0.05
0.12
0.1
0.03
-
0.02
0.02
)
,
S
=
(
0.6
0.2
0.1
0.2
0.6
0.4
0.1
0.4
0.2
)
,
S
1
=
(
0.4
-
0.1
0.3
-
0.1
0.3
0.2
0.3
0.2
0.6
)
,
S
2
=
(
0.5
0.1
0.2
0.1
0.3
0.4
0.2
0.4
0.6
)
,
R
=
(
1.12
0.3
-
0.1
0.3
1.5
0
-
0.1
0
1.6
)
,
L
1
=
(
-
0.35
0.2
0.1
0.2
0.3
0.1
0.8
0.2
0.1
)
,
L
2
=
(
1.2
0.23
2
-
1.3
0.4
0.2
1.5
-
0.3
1.6
)
,
d
(
k
)
=
1.5
+
1.5
sin
k
π
2
,
h
(
k
)
=
1
+
cos
⁡
k
π
2
,
k
=
0,1
,
2
,
…
.
Given α
=
0.032, d
=
3, h
=
2, β
1
=
0.23, β
2
=
0.25, and the initial condition
(48)
x
(
-
3
)
=
(
-
1.6
1.3
1.2
)
,
x
(
-
2
)
=
(
-
1.3
0.8
-
1
)
,
x
(
-
1
)
=
(
0.4
-
0.5
0.2
)
,
x
(
0
)
=
(
1.2
0
-
1.2
)
,
using the LMI Toolbox in MATLAB [14], the LMI (10) in Theorem 5 is satisfied with
(49)
P
=
(
4.1184
0.0455
-
0.3818
0.0455
1.6976
-
0.2973
-
0.3818
-
0.2973
10.3666
)
,
Q
1
=
(
0.0311
-
0.0020
-
0.0207
-
0.0020
0.0115
-
0.0158
-
0.0207
-
0.0158
0.1978
)
,
Q
2
=
(
0.0478
-
0.0032
-
0.0334
-
0.0032
0.0163
-
0.0254
-
0.0334
-
0.0254
0.3176
)
,
R
1
=
(
0.1001
-
0.0069
-
0.0710
-
0.0069
0.0329
-
0.0541
-
0.0710
-
0.0541
0.6715
)
,
R
2
=
(
0.1046
-
0.0077
-
0.0783
-
0.0077
0.0299
-
0.0596
-
0.0783
-
0.0596
0.7340
)
,
N
=
(
5.2270
0.1211
0.0022
0.1220
2.2186
-
0.4808
0.0003
-
0.4800
13.0762
)
,
ε
1
=
7.3374
,
ε
2
=
6.7127
,
and the controller parameter
(50)
K
=
(
0.6351
-
0.9829
-
0.4757
-
0.9829
2.7965
0.8749
-
0.4757
0.8749
0.9870
)
.
Moveover, the solution of the closed-loop system satisfies
(51)
∥
x
(
k
)
∥
≤
4.0345
e
-
0.032
k
∥
ϕ
∥
,
and the guaranteed cost of the closed-loop system is as follows
(52)
J
*
=
27.4600
∥
ϕ
∥
2
.
Simulation result is presented in Figure 1, which shows the convergence behavior of the proposed methods.Figure 1
State trajectories of the closed-loop system.


## 5. Conclusion

In this paper, the problem of robust output feedback guaranteed cost control for nonlinear uncertain disctere system is researched. For all admissible uncertainties, an output feedback guaranteed cost controller has been designed such that the resulting closed-loop system is robust exponentially stable and guarantees an adequate level of system performance. A numerical example has been presented to illustrate the efficiency of the result.



---

*Source: 628041-2014-09-10.xml*