# H

Authors: Ting Hou; Hongji Ma
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

An infinite horizonH
2
/
H
∞ control problem is addressed for discrete-time periodic Markov jump systems with (
x
,
u
,
v
)-dependent noise. Above all, by use of the spectral criterion of detectability, an extended Lyapunov stability theorem is developed for the concerned dynamics. Further, based on a game theoretic approach, a state-feedback H
2
/
H
∞ control design is proposed. It is shown that under the condition of detectability H
2
/
H
∞ feedback gain can be constructed through the solution of a group of coupled periodic difference equations.


---

## Body

## 1. Introduction

H
∞ control has been one of the most active areas of modern control theory since the 1970s. Owing to the introduction of state-space approach [1], many researchers have been inspired to extend the deterministic H
∞ control theory to various stochastic systems; see [2–5]. In the development of stochastic H
∞ theory, [2] can be regarded as a pioneering work, which firstly established a stochastic version of bounded real lemma for linear Itô-type differential systems. Besides, initialed from [6], considerable progress has been made in the study of stochastic H
2
/
H
∞ control. By combing H
∞ index with an quadratic cost performance, the resulting multiobjective control strategy is more attractive than the sole H
∞ control in engineering applications.The main objective of this paper is to settle an infinite horizonH
2
/
H
∞ control problem for periodic Markov jump systems with multiplicative noises. By now, Markov jump systems have been extensively investigated [7–9]. For example, stochastic and robust stability have been elaborately discussed in [10, 11] for networked dynamics with Markovian jump. As concerns H
∞ theory, an H
∞ estimation problem was tackled in [12] for a class of discrete homogeneous Markov jump systems. On the other hand, an infinite horizon H
∞ control problem was handled in [13] for nonlinear Itô systems with homogeneous Markov process. However, few results have been reported for H
2
/
H
∞ control of periodic Markov jump systems. To some extent, this study will generalize the work of [14] to the case of periodically time-varying coefficients and transition probabilities, as in [15–17].The remainder of this paper is organized as follows. Section2 gives basic preliminaries and problem formulations. In Section 3, the intrinsic relationship between asymptotic mean square stability and detectability is addressed. As a result, a Barbashin-Krasovskii-type theorem is established for periodic Markov jump systems with state-dependent noises. Section 4 contains an internally stabilizing control design, which can not only fulfill the prescribed disturbance attenuation level, but also minimize the output energy. To verify the effectiveness of the proposed approach, a numerical example is supplied in Section 5. Finally, Section 6 concludes this paper with a concluding remark.Notations. R
n (C
n) is n-dimensional real (complex) space with the usual Euclidean norm ·; R
n
×
m is the space of all n
×
m real matrices with the operator norm ·
2; S
n is the set of all n
×
n symmetric matrices whose entries may be complex; A
>
0
(
≥
0
): A
∈
S
n is a positive (semi)definite matrix; A
′ is the transpose of a matrix (vector) A; I
n is the n
×
n identity matrix; Z
+
≔
{
0,1
,
…
} and Z
1
+
≔
{
1,2
,
…
}; D
=
{
1,2
,
…
,
N
}; ⊗ is the operation of Kronecker product; K
e
r
(
·
) is the kernel of a matrix; d
i
a
g
{
·
} is a (block) diagonal matrix.


## 2. Preliminaries

On a complete probability space(
Ω
,
F
,
P
), we consider the following discrete-time periodic Markov jump system with (
x
,
u
,
v
)-dependent noise:(1)
x
t
+
1
=
A
0
t
,
η
t
x
t
+
B
0
t
,
η
t
v
t
+
G
0
t
,
η
t
u
t
+
∑
s
=
1
r
A
s
t
,
η
t
x
t
+
B
s
t
,
η
t
v
t
+
G
s
t
,
η
t
u
t
w
s
t
,
z
t
=
C
t
,
η
t
x
t
D
t
,
η
t
u
t
,
D
t
,
η
t
′
D
t
,
η
t
=
I
n
u
,
t
∈
Z
+
, where x
(
t
)
∈
R
n, u
(
t
)
∈
R
n
u, v
(
t
)
∈
R
n
v, and z
(
t
)
∈
R
n
z denote the system state, control input, exogenous disturbance, and measurement output, respectively. Assume that w
t
∣
w
t
=
w
1
t
,
…
,
w
r
t
′
,
t
∈
Z
+ is a sequence of independent random vectors such that E
[
w
(
t
)
]
=
0 and E
[
w
t
w
s
′
]
=
I
r
δ
(
t
-
s
) (δ
(
·
) is a Kronecker function). The Markov chain {
η
t
}
t
∈
Z
+ takes values in D with a nondegenerate transition probability matrix P
t
=
[
p
t
(
i
,
j
)
]
N
×
N
(
p
t
(
i
,
j
)
≔
P
η
t
+
1
=
j
∣
η
t
=
i
) and the initial distribution π
0
(
i
)
=
P
(
η
0
=
i
)
>
0 for all i
∈
D. As usual, we set that {
η
t
}
t
∈
Z
+ is independent of the stochastic process {
w
(
t
)
}
t
∈
Z
+ and its mode is measurable in real time. Moreover, the coefficients of (1) are θ-periodic (e.g., A
s
(
t
,
i
)
=
A
s
(
t
+
θ
,
i
)) and the transition probability of η
t satisfies p
t
(
i
,
j
)
=
p
t
+
θ
(
i
,
j
)
(
i
,
j
∈
D
), where θ
∈
Z
1
+. Let F
k be σ-algebra generated by η
t
,
w
s
∣
0
≤
t
≤
k
,
0
≤
s
≤
k
-
1. In the case of k
=
0,  F
0
=
σ
{
η
0
}. Denote by l
2
(
0
,
∞
;
R
m
) the space of R
m-valued, nonanticipative square summable stochastic processes {
y
(
t
,
ω
)
:
Z
+
×
Ω
→
R
m
} which are F
k-measurable for all k
∈
Z
+ and ∑
t
=
0
∞
E
y
(
t
)
2
<
+
∞. It is clear that l
2
(
0
,
∞
;
R
m
) is a real Hilbert space with the norm induced by the usual inner product: y
(
·
)
l
2
(
0
,
∞
;
R
m
)
=
(
∑
t
=
0
∞
E
y
(
t
)
2
)
1
/
2
<
+
∞.Definition 1 (see [18]).
The zero-state equilibrium of discrete-time periodic Markov jump systems(2)
x
t
+
1
=
A
t
,
η
t
x
t
+
∑
s
=
1
r
A
s
t
,
η
t
x
t
w
s
t
,or (
A
;
P
) is called asymptotically mean square stable (AMSS) if l
i
m
t
→
∞
E
x
(
t
;
ξ
,
i
,
t
0
)
2
=
0 for all (
ξ
,
i
)
∈
R
n
×
D and t
0
∈
Z
+. Here, x
(
t
;
ξ
,
i
,
t
0
) is the state of (2) corresponding to the initial state x
(
t
0
)
=
ξ
∈
R
n and η
t
0
=
i
∈
D. Moreover, if there exists θ-periodic sequence {
K
(
t
,
i
)
}
t
∈
Z
+
∈
R
n
×
n
u
(
i
∈
D
) such that the zero-state equilibrium of the closed-loop system(3)
x
t
+
1
=
A
0
t
,
η
t
+
G
0
t
,
η
t
K
t
,
η
t
x
t
+
∑
s
=
1
r
A
s
t
,
η
t
+
G
s
t
,
η
t
K
t
,
η
t
x
t
w
s
tis AMSS for any (
x
0
,
η
0
)
∈
R
n
×
D, then (
A
,
G
;
P
) is called stochastically stabilizable and u
(
t
)
=
K
(
t
,
η
t
)
x
(
t
) is called a stabilizing feedback.By Theorem 3.10 [18], we know that system (2) is asymptotically mean square stable if and only if it is exponentially mean square stable.Definition 2 (see [19]).
The periodic Markov jump system with measurement equation(4)
x
t
+
1
=
A
0
t
,
η
t
x
t
+
∑
s
=
1
r
A
s
t
,
η
t
x
t
w
s
t
,
z
t
=
C
t
,
η
t
x
t
,
t
∈
Z
+
, or (
A
,
C
;
P
) is called (uniformly) detectable if for any t
0
∈
Z
+, ξ
∈
R
n, and η
t
0
∈
D, there holds(5)
z
t
≡
0
a.s.
,
t
∈
t
0
,
T
,
∀
T
>
t
0
⟹
lim
t
→
∞
⁡
E
x
t
;
ξ
,
i
,
t
0
2
=
0
.In this paper, we will deal with the infinite horizon optimalH
2
/
H
∞ control problem about (1). More specifically, for a prescribed disturbance attenuation level γ
>
0, we aim to find a linear, memoryless, periodic state-feedback controller u
∗
(
t
)
∈
l
2
(
0
,
∞
;
R
n
u
) such that [20](i)
whenv
(
t
)
≡
0, the closed-loop state of (1) corresponding to u
=
u
∗
(
t
) is AMSS;
(ii)
thel
2-induced norm of L
∞
u
∗ satisfies L
∞
u
∗
<
γ, where L
∞
u
∗ is the perturbation operator defined by L
∞
u
∗
(
v
)
=
z
(
t
;
0
,
v
,
u
∗
); it is notable that z
(
t
;
0
,
v
,
u
∗
) is the output of (1) corresponding to x
(
0
)
=
0 and u
=
u
∗
(
t
), while v
(
t
)
∈
l
2
(
0
,
∞
;
R
n
v
) is arbitrary random disturbance;
(iii)
when the worst-case disturbancev
∗
(
t
), if existing, is imposed on (1), u
∗
(
t
) minimizes the corresponding output energy J
2
(
u
,
v
∗
)
≔
∑
t
=
0
∞
E
z
(
t
)
2.


## 3. Stability and Detectability

In this section, we will focus on the detectability of periodic Markov jump system (1). This structural property will play an essential role in the treatment of H
2
/
H
∞ control problem. Firstly, we present several instrumental operators.LetS
n
N (resp., S
n
N
+) indicate the set of all N sequences V
=
(
V
(
1
)
,
…
,
V
(
N
)
) with V
(
i
)
∈
S
n (resp., V
(
i
)
≥
0). Thus, S
n
N is a Hilbert space with the inner product:(6)
U
,
V
=
∑
i
=
1
N
Tr
⁡
U
i
V
i
,
∀
U
,
V
∈
S
n
N
.GivenU
∈
S
n
N, let L
t
:
S
n
N
→
S
n
N be a Lyapunov operator defined as L
t
(
U
)
=
(
L
t
(
U
,
1
)
,
…
,
L
t
(
U
,
N
)
), where(7)
L
t
U
,
i
=
∑
s
=
0
r
∑
j
=
1
N
p
t
j
,
i
A
s
t
,
j
U
j
A
s
t
,
j
′
.Then, associated with inner product (6), the adjoint operator of L
t is given by L
t
∗
(
U
)
=
(
L
t
∗
(
U
,
1
)
,
…
,
L
t
∗
(
U
,
N
)
):(8)
L
t
∗
U
,
i
=
∑
s
=
0
r
A
s
t
,
i
′
∑
j
=
1
N
p
t
i
,
j
U
j
A
s
t
,
i
.In terms ofL
t, we can construct a causal evolution T
t
,
s
=
L
t
-
1
⋯
L
s
(
t
>
s
≥
0
); when t
=
s, T
t
,
t
=
I (i.e., the identity operator).To proceed, let us introduce the following two linear operators (cf. [14, 21]):(9)
ψ
X
=
ψ
X
1
′
⋯
ψ
X
N
′
′
,
φ
X
=
φ
X
1
′
⋯
φ
X
N
′
′
,
ψ
X
i
=
x
11
i
⋯
x
1
n
i
⋯
x
n
1
i
⋯
x
n
n
i
′
,
φ
X
i
=
x
11
i
⋯
x
1
n
i
x
22
i
⋯
x
2
n
i
⋯
x
n
-
1
,
n
-
1
i
x
n
-
1
,
n
i
x
n
n
i
′
,where X
∈
S
n
N and x
j
k
(
i
) is the entry of X
(
i
). It is easy to verify that ψ
:
S
n
N
→
C
n
2
N and φ
:
S
n
N
→
C
n
(
n
+
1
)
/
2
N are both invertible and satisfy (10)
ψ
L
t
X
=
M
t
ψ
X
,
φ
L
t
X
=
L
t
φ
X
,
ψ
X
=
H
φ
X
,
∀
X
∈
S
n
N
,where H
∈
R
n
2
N
×
n
(
n
+
1
)
/
2
N is a constant matrix of full column rank and(11)
M
t
=
P
t
′
⊗
I
n
2
Δ
∑
s
=
1
r
A
s
t
,
i
⊗
A
s
t
,
i
,
L
t
=
H
′
H
-
1
H
M
t
H
.In (11), L
t is called the induced matrix of L
t and Δ
{
A
(
i
)
}
≔
d
i
a
g
{
A
(
1
)
,
…
,
A
(
N
)
} (if A
(
i
)
≡
A for i
∈
D, then Δ
(
A
)
≔
d
i
a
g
{
A
,
…
,
A
︸
N
}). Repeating the above steps, the induced matrix of T
t
,
s is realized to be A
t
,
s
=
L
t
-
1
⋯
L
s. Particularly, the induced matrix of T
t
θ is denoted by A
t
θ
≔
A
t
+
θ
,
t.Next, we will give two useful lemmas, which have been shown in [19].Lemma 3.
(
A
;
P
) is AMSS if and only if Λ
(
T
t
θ
)
=
Λ
(
A
t
θ
)
⊂
⊙, where Λ
(
·
) denotes the spectral set of an operator (or a matrix) and ⊙
≔
{
λ
∈
C
∣
λ
<
1
}.Lemma 4.
(
A
,
C
;
P
) is detectable if and only if for some t
∈
Z
+, there does not exist any nonzero X
∈
S
n
N such that(12)
T
t
θ
X
,
i
=
λ
X
i
,
λ
∈
C
,
λ
≥
1
,
i
∈
D
,
C
j
,
i
T
j
,
t
X
,
i
=
0
,
j
=
t
,
…
,
t
+
θ
-
1
.We are prepared to establish the following Barbashin-Krasovskii stability criterion for (4).Theorem 5.
If(
A
,
C
;
P
) is detectable, then (
A
;
P
) is AMSS if and only if the PLE(13)
X
t
i
=
L
t
∗
X
t
+
1
,
i
+
C
t
,
i
′
C
t
,
i
,
t
∈
Z
+has a unique θ-periodic solution X
t
∈
S
n
N
+.Proof.
By Theorem 2.5 [18], if (
A
;
P
) is AMSS, then the PLE (13) admits a unique θ-periodic solution X
t
∈
S
n
N
+. Next, we will show the converse assertion. If (13) has θ-periodic solution X
t
∈
S
n
N
+ but (
A
;
P
) is not AMSS, by Lemma 3, there must exist λ
∈
Λ
(
T
t
θ
) with |
λ
|
≥
1. Denote by ρ
(
T
t
θ
) the spectral radius of T
t
θ; then ρ
(
T
t
θ
)
≥
1. According to the Krein-Rutman theorem, there is a positive definite X
¯
∈
S
n
N such that T
t
θ
(
X
¯
)
=
ρ
(
T
t
θ
)
X
¯. Since (
A
,
C
;
P
) is detectable, by Lemma 4, for some t
∈
Z
+, there exists at least one k
∈
{
t
,
…
,
t
+
θ
-
1
} such that(14)
C
k
T
k
,
t
X
¯
≠
0
.Thus, for inner product (6), it can be computed from (13) that(15)
0
≤
∑
k
=
t
t
+
θ
-
1
∑
i
=
1
N
Tr
⁡
C
k
,
i
T
k
,
t
X
¯
,
i
C
k
,
i
′
=
∑
k
=
t
t
+
θ
-
1
C
k
′
C
k
,
T
k
,
t
X
¯
=
X
t
,
X
¯
-
X
t
+
θ
,
T
t
θ
X
¯
.Due to the periodicity of X
t, (15) leads to the fact that (16)
0
≤
∑
k
=
t
t
+
θ
-
1
∑
i
=
1
N
Tr
⁡
C
k
,
i
T
k
,
t
X
¯
,
i
C
k
,
i
′
=
X
t
,
X
¯
-
T
t
θ
X
¯
=
X
t
,
1
-
ρ
T
t
θ
X
¯
≤
0
,which implies C
k
,
i
T
k
,
t
X
¯
,
i
C
k
,
i
′
=
0 for t
≤
k
≤
t
+
θ
-
1 and i
∈
D. That is, C
(
k
)
T
k
,
t
(
X
¯
)
=
0 for t
≤
k
≤
t
+
θ
-
1, which contradicts (14). Hence, (
A
;
P
) is AMSS.Remark 6.
In [18], a similar result has been proven under the condition of stochastic detectability. According to [19], (uniform) detectability is a weaker prerequisite than stochastic detectability. Therefore, Theorem 5 has improved the result of Theorem 4.1 [18] within the concerned framework.


## 4.H
2
/
H
∞ Control

In this section, a game theoretic approach will be employed to deal with the infinite horizonH
2
/
H
∞ control problem of (1). Under the assumption of detectability, a necessary and sufficient condition can be provided for the existence of H
2
/
H
∞ controller.Theorem 7.
For system (1), if the following coupled periodic difference equations (CPDEs) admit θ-periodic quaternion solution (
X
1
(
t
), F
1
(
t
); X
2
(
t
), F
2
(
t
)
)
∈
S
n
N
+
×
R
n
×
n
v
N
×
S
n
N
+
×
R
n
×
n
u
N on Z
+
×
D,(17)
X
1
t
,
i
=
∑
s
=
0
r
A
¯
s
2
t
,
i
′
E
t
X
1
t
+
1
,
i
A
¯
s
2
t
,
i
+
C
t
,
i
′
C
t
,
i
+
F
2
t
,
i
′
F
2
t
,
i
+
F
3
t
,
i
H
1
t
,
i
-
1
F
3
t
,
i
′
,
H
1
t
,
i
≥
ɛ
0
I
n
v
,
ɛ
0
∈
0
,
γ
2
-
L
∞
u
∗
2
,
(18)
F
1
t
,
i
=
H
1
t
,
i
-
1
F
3
t
,
i
′
,
(19)
X
2
t
,
i
=
∑
s
=
0
r
A
¯
s
1
t
,
i
′
E
t
X
2
t
+
1
,
i
A
¯
s
1
t
,
i
+
C
t
,
i
′
C
t
,
i
-
F
4
t
,
i
H
2
t
,
i
-
1
F
4
t
,
i
′
,
(20)
F
2
t
,
i
=
-
H
2
t
,
i
-
1
F
4
t
,
i
′
,where (21)
A
¯
s
1
t
,
i
=
A
s
t
,
i
+
B
s
t
,
i
F
1
t
,
i
,
A
¯
s
2
t
,
i
=
A
s
t
,
i
+
G
s
t
,
i
F
2
t
,
i
,
H
1
t
,
i
=
γ
2
I
n
v
-
∑
s
=
0
r
B
s
t
,
i
′
E
t
X
1
t
+
1
,
i
B
s
t
,
i
,
H
2
t
,
i
=
I
n
u
+
∑
s
=
0
r
G
s
t
,
i
′
E
t
X
2
t
+
1
,
i
G
s
t
,
i
,
F
3
t
,
i
=
∑
s
=
0
r
A
¯
s
2
t
,
i
′
E
t
X
1
t
+
1
,
i
B
s
t
,
i
,
F
4
t
,
i
=
∑
s
=
0
r
A
¯
s
1
t
,
i
′
E
t
X
2
t
+
1
,
i
G
s
t
,
i
,and (
A
,
C
;
P
), (
A
+
B
F
1
,
C
;
P
) are detectable, then the state-feedback H
2
/
H
∞ control is given by (
u
∗
(
t
)
=
F
2
(
t
,
η
t
)
x
(
t
)
,
v
∗
(
t
)
=
F
1
(
t
,
η
t
)
x
(
t
)
).
Conversely, if(
A
+
B
F
1
,
C
;
P
) is detectable and the H
2
/
H
∞ control problem about (1) is solved by (
u
∗
(
t
)
=
F
2
(
t
,
η
t
)
x
(
t
)
,
v
∗
(
t
)
=
F
1
(
t
,
η
t
)
x
(
t
)
), then CPDEs (17)–(20) admit a unique θ-periodic quaternion solution (
X
1
(
t
), F
1
(
t
); X
2
(
t
), F
2
(
t
)
)
∈
S
n
N
+
×
R
n
×
n
v
N
×
S
n
N
+
×
R
n
×
n
u
N on Z
+
×
D.Proof.
“⇒”: (a) Let us first show that u
∗ stabilizes system (1) internally (
v
(
t
)
≡
0
). To this end, we rewrite (19) as follows:(22)
X
2
t
,
i
=
∑
s
=
0
r
A
¯
s
1
t
,
i
+
G
s
t
,
i
F
2
t
,
i
′
E
t
X
1
t
+
1
,
i
A
¯
s
1
t
,
i
+
G
s
t
,
i
F
2
t
,
i
+
C
1
t
,
i
′
C
1
t
,
i
,where (23)
C
1
t
,
i
=
C
t
,
i
H
1
t
,
i
-
1
/
2
F
3
t
,
i
F
2
t
,
i
.Since (
A
+
B
F
1
,
C
;
P
) is detectable, by Lemma 4, we can prove that (
A
+
B
F
1
+
G
F
2
,
C
1
;
P
) is also detectable. From (22) and Theorem 5, it follows that (
A
+
B
F
1
+
G
F
2
;
P
) is AMSS, which means u
∗
(
t
)
=
F
2
(
r
t
)
x
(
t
)
∈
l
2
(
0
,
∞
;
R
n
u
) and v
∗
(
t
)
=
F
1
(
r
t
)
x
(
t
)
∈
l
2
(
0
,
∞
;
R
n
v
). Similarly, we can prove that (
A
+
G
F
2
;
P
) is also AMSS. Hence, u
∗
(
t
) can stabilize system (1) internally.
(b) Consider the following:L
∞
u
∗
<
γ. Implementing u
∗
(
t
)
=
F
2
(
t
,
η
t
)
x
(
t
) into system (1), we get(24)
x
t
+
1
=
A
¯
0
2
t
,
η
t
x
t
+
B
0
t
,
η
t
v
t
+
∑
s
=
1
r
A
¯
s
1
t
,
η
t
x
t
+
B
s
t
,
η
t
v
t
w
s
t
,
z
t
=
C
t
,
η
t
D
t
,
η
t
F
2
t
,
η
t
x
t
,
D
t
,
η
t
′
D
t
,
η
t
=
I
n
u
,
t
∈
Z
+
.Noting that (
A
+
B
F
1
;
P
) is AMSS, X
1
∈
S
n
N
+ is a stabilizing solution of (17). By use of Theorem 1 [20], we deduce that system (24) satisfies L
∞
u
∗
<
γ.
(c)u
∗ minimizes the performance J
2
(
u
,
v
∗
). By (17) and (24), we can complete square as follows:(25)
J
∞
u
∗
,
v
≔
∑
t
=
0
∞
E
z
t
2
-
γ
2
v
t
2
=
∑
i
=
1
N
π
0
i
x
0
′
X
1
0
,
i
x
0
-
∑
t
=
0
∞
E
v
t
-
H
1
t
,
η
t
-
1
F
3
t
,
η
t
′
x
t
′
H
1
t
,
η
t
v
t
-
H
1
t
,
η
t
-
1
F
3
t
,
η
t
′
x
t
≤
∑
i
=
1
N
π
0
i
x
0
′
X
1
0
,
i
x
0
=
J
∞
u
∗
,
v
∗
,which implies that v
∗
(
t
)
=
F
1
(
t
,
η
t
)
x
(
t
) is the worst-case disturbance associated with u
∗. Applying v
∗ to system (1), we have(26)
x
t
+
1
=
A
¯
0
1
t
,
η
t
x
t
+
G
0
t
,
η
t
u
t
+
∑
s
=
1
r
A
¯
s
1
t
,
η
t
x
t
+
G
s
t
,
η
t
u
t
w
s
t
,
z
t
=
C
t
,
η
t
x
t
D
t
,
η
t
u
t
,
D
t
,
η
t
′
D
t
,
η
t
=
I
n
u
,
t
∈
Z
+
.
It remains to show thatu
∗ fulfills the following optimal index:(27)
m
i
n
u
·
∈
l
2
0
,
∞
;
R
n
u
J
2
u
,
v
∗
=
∑
t
=
0
∞
E
z
t
2
,
subject to
26
,which is an LQ optimal control problem. Since (19) is equivalent to (22) and (
A
+
B
F
1
+
G
F
2
,
C
1
;
P
) is detectable, by Theorem 5, X
2
∈
S
n
N
+ is a stabilizing solution of (19). Making use of Proposition 6.3 [18], we arrive at(28)
J
2
u
,
v
∗
=
∑
i
=
1
N
π
0
i
x
0
′
X
2
0
,
i
x
0
+
∑
t
=
0
∞
E
u
t
+
H
2
t
,
η
t
-
1
F
4
t
,
η
t
′
x
t
′
H
2
t
,
η
t
u
t
+
H
2
t
,
η
t
-
1
F
4
t
,
η
t
′
x
t
≥
∑
i
=
1
N
π
0
i
x
0
′
X
2
0
,
i
x
0
=
J
2
u
∗
,
v
∗
,where u
∗
=
F
2
(
t
,
η
t
)
x
(
t
). This justifies the sufficiency statement.
“⇐”: Assume that (
u
∗
(
t
)
=
F
2
(
t
,
η
t
)
x
(
t
) and v
∗
(
t
)
=
F
1
(
t
,
η
t
)
x
(
t
)
) solve the considered H
2
/
H
∞ control problem. Thus, u
∗ stabilizes system (1) internally and L
∞
u
∗
<
γ. By Theorem 1 [20], we conclude that (17) admits a stabilizing solution X
1
∈
S
n
N
+, which implies that (
A
+
B
F
1
+
G
F
2
;
P
) is AMSS. Since system (24) is internally stable, by Corollary 3.9 [18], we deduce that x
(
t
)
∈
l
2
(
0
,
∞
;
R
n
) for any v
(
t
)
∈
l
2
(
0
,
∞
;
R
n
v
). As shown in the sufficient part, by use of (17) and (24), we will come to (25), which indicates F
1
(
t
,
η
t
)
=
H
1
(
t
,
η
t
)
-
1
F
3
t
,
η
t
′. Further, imposing v
∗ on system (1) gives (26). Because u
∗ is the optimal H
2
/
H
∞ control, u
∗ solves LQ control problem (27). Moreover, from detectability of (
A
+
B
F
1
,
C
;
P
), we have that (
A
+
B
F
1
+
G
F
2
,
C
1
;
P
) is detectable. Recalling that (
A
+
B
F
1
+
G
F
2
;
P
) is AMSS, by Theorem 5, (22) (i.e., (19)) has a stabilizing solution X
2
∈
S
n
N
+. Finally, by completing square in terms of (19) and (26), we obtain (28), which justifies that F
2
(
t
,
η
t
)
=
-
H
2
(
t
,
η
t
)
-
1
F
4
t
,
η
t
′. This ends the proof.Remark 8.
If the coefficients of (1) reduce to be time-invariant and the Markov chain {
η
t
}
t
∈
Z
+ is homogeneous, then Theorem 7 is reduced to the conclusion of Theorem 3 [14]. Hence, the current study can be regarded as a periodic extension of [14]. At present, there still exists some difficulty in generalizing the above method to design H
2
/
H
∞ controller for time-varying Markov jump systems as in [22]. To this end, a time-varying version of PBH criterion has to be developed.


## 5. Numerical Example

Consider the following two-dimensional Markov jump system with the periodic coefficients listed as follows:(29)
A
0
t
,
1
=
0.88
-
0.35
cos
⁡
π
t
0
0
0.6
,
A
0
t
,
2
=
0.66
0
0
0.73
,
B
0
t
,
1
=
0.52
0.31
,
B
0
t
,
2
=
0.76
0.53
,
G
0
t
,
1
=
1.4
+
0.2
cos
⁡
π
t
0.49
,
G
0
t
,
2
=
1.1
1.04
,
A
1
t
,
1
=
0.63
0
0
0.72
,
A
1
t
,
2
=
0.48
-
0.2
cos
⁡
π
t
0
0.6
,
B
1
t
,
1
=
0.4
-
0.1
cos
⁡
π
t
0.5
,
B
1
t
,
2
=
0.1
0.4
,
G
1
t
,
1
=
1.1
1.04
,
G
1
t
,
2
=
0.4
0.3
. Moreover, C
t
,
i
≡
0.9
0.56 and D
(
t
,
i
)
≡
1. It is clear that θ
=
2. The transition probability matrix P
t is determined by p
2
k
(
1,1
)
=
p
2
k
(
2,1
)
=
0.2, p
2
k
(
1,2
)
=
p
2
k
(
2,2
)
=
0.8, p
2
k
+
1
(
1,1
)
=
p
2
k
+
1
(
2,1
)
=
0.3, and p
2
k
+
1
(
1,2
)
=
p
2
k
+
1
(
2,2
)
=
0.7
(
k
∈
Z
+
). For a prescribed disturbance attenuation level γ
=
2.3, by use of the Runge-Kutta algorithm, we can solve CPDEs (17)–(20) and get the feedback gains of (
u
∗
,
v
∗
): (30)
F
1
2
k
,
1
=
0.12
0.024
,
F
1
2
k
,
2
=
0.03
0.05
,
F
2
2
k
,
1
=
-
0.49
-
0.21
,
F
2
2
k
,
2
=
-
0.34
-
0.27
;
F
1
2
k
+
1,1
=
-
0.06
0.05
,
F
1
2
k
+
1,2
=
0.04
0.016
,
F
2
2
k
+
1,1
=
-
0.38
-
0.235
,
F
2
2
k
+
1,2
=
-
0.35
-
0.25
,
k
∈
Z
+
.By Lemma 4, it can be verified that (
A
,
C
;
P
) and (
A
+
B
F
1
,
C
;
P
) are both detectable. Applying (
u
∗
,
v
∗
) to the periodic Markov jump system, we get the closed-loop state trajectory and corresponding H
2 performance. Figure 1(a) has displayed 50 sampled state trajectories originating from (
x
1
(
0
)
,
x
2
(
0
)
)
=
(
10,20
), while Figure 1(b) demonstrates the cumulative energy of the system output.Figure 1
State responses andH
2 performance.
(a)
(b)


## 6. Conclusion

In this paper, an infinite horizonH
2
/
H
∞ control problem has been settled for discrete-time periodic Markov jump systems with multiplicative noise. Under the condition of (uniform) detectability, a game theoretic H
2
/
H
∞ control is produced by solving a group of CPDEs. Note that there remain some open topics on this issue. For example, it is interesting as well as challenging to investigate the H
2
/
H
∞ control problem with input or output saturation constraint [23], which no doubt deserves a further study.



---

*Source: 382756-2015-10-19.xml*