# Static Output Feedback Control for Discrete-Time Switched Systems via Improved Path-Following Method

Authors: Jian Chen; Chong Lin
Journal: Discrete Dynamics in Nature and Society
 (2015)

---

## Abstract

This paper focuses on the problems of static output feedback control andH
∞ controller design for discrete-time switched systems. Based on piecewise quadratic Lyapunov functions and a new linearization method, new sufficient conditions for system stability and H
∞ controller design are obtained. Then, an improved path-following algorithm is built to solve the problems. Finally, the merits and effectiveness of the proposed method are shown by two numerical examples.


---

## Body

## 1. Introduction

Recently, there has been an increasing interest in the study of switched systems because a wide class of nonlinear systems are naturally written as switched systems [1]. Moreover, many other types of nonlinear systems can also be modeled as switched systems approximately [2]. Switched systems are a particular class of hybrid systems which are bounded together by a switching rule. Such systems can be used to describe a wide range of physical and engineering systems in practice [3].There are a large number of literatures about the stability analysis and design of switched systems during the last few years [4–9]. For discrete-time systems, several sufficient conditions have been presented based on different Lyapunov functions [6, 7, 10, 11], which are different in the conservative level and in the numerical difficulties. Previous work has concentrated on the output feedback controller design methods for switched systems based on piecewise quadratic Lyapunov function [12, 13]. Lyapunov-based controller synthesis is formulated as a biconvex optimization problem which is nonconvex, NP-hard, and very expensive to solve globally [14]. Although there exist some results in solving this problem that the corresponding conditions can be determined by checking a set of linear matrix inequalities (LMIs) [12, 13], most of them are very restrictive.Path-following method, which is an effective method for solving the biconvex optimization problem, was proposed by Hassibi et al. [15] and employed to solve mixed H
2
/
H
∞ control [16, 17] and other control problems [18]. An improved path-following method [19] has enhanced the convergence and the performance of the algorithm. As a step-by-step method, implying linearization approach at its key step, it gradually shows enormous potential in solving control problems.In this paper, the problem of static output feedback (SOF) control for discrete-time switched systems is studied. Based on piecewise quadratic Lyapunov functions [12, 13] and a new linearization method [19], the piecewise quadratic stability conditions are linearized around some points. As a result, less conservative conditions for system stability are derived. The problems of H
∞ control design can be readily treated as well. Then, based on an improved path-following method, an iterative algorithm is built. Finally, two examples are given to show the merits and effectiveness of our work.This paper is organized as follows. Section2 is the problem formulation and preliminaries. Section 3 gives the SOF controller design for switched systems. Section 4 extends the method to H
∞ SOF control. Section 5 provides two numerical examples to show the merits and effectiveness of the results and Section 6 concludes this paper.Notation. R
n denotes the n-dimensional Euclidean space; the superscripts -
1 and T denote the matrix inverse and transpose, respectively; X
>
0
(
X
≥
0
) means that X is positive definite (positive semi-definite); · is the spectral norm; the star ∗ denotes the symmetric term in a matrix; l
(
X
,
Y
)
=
X
Y
+
X
T
Y
T.


## 2. Problem Formulation and Preliminaries

Consider the discrete-time switched system:(1)
x
k
+
1
=
A
i
x
k
+
B
i
u
k
,
y
k
=
C
i
x
k
,
for
x
∈
S
i
,
i
=
1,2
,
…
,
l
,where x
(
k
)
∈
R
n, u
(
k
)
∈
R
m, and y
(
k
)
∈
R
p are the state, the control input, and the measured output, respectively; S
i
⊆
R
n, i
=
1,2
,
…
,
l, denotes a partition of the output space into a number of closed polyhedral regions. For future use, define a set Ω to represent all possible switches from one region to itself or another region; that is(2)
Ω
=
i
,
j
:
i
,
j
=
1,2
,
…
,
l
,
s.t.
x
k
∈
S
i
,
x
k
+
1
∈
S
j
.The set Ω can be determined by the reachability analysis for mixed logical dynamical systems. The system is allowed to switch arbitrarily between subsystems.We study the problem of designing a static output feedback controller:(3)
u
k
=
F
i
y
k
,
i
=
1,2
,
…
,
l
,where F
i
∈
R
m
×
p such that the closed-loop switched system(4)
x
k
+
1
=
A
i
+
B
i
F
i
C
i
x
k
,
i
=
1,2
,
…
,
lis stable.The following lemmas give the stability condition of closed-loop systems (4) and the new linearization method proposed in [19].Lemma 1 (see [12]).
If there exist matricesP
i
=
P
i
T
>
0  (
∀
i
=
1,2
,
…
,
l
), such that the positive definite function V
(
x
)
=
x
T
P
i
x
(
∀
x
∈
S
i), satisfies V
(
x
(
k
+
1
)
)
-
V
(
x
(
k
)
)
<
0, that is,(5)
A
i
+
B
i
F
i
C
i
T
P
j
A
i
+
B
i
F
i
C
i
-
P
i
<
0
,
∀
i
,
j
∈
Ω
,then the closed-loop switched systems (4) are exponentially stable.Lemma 2 (see [19]).
If there exists a fixed point(
M
,
N
) such that the LMI(6)
l
M
+
Δ
A
,
N
+
l
M
,
Δ
P
∗
Δ
A
T
+
Δ
P
-
2
I
<
0holds for some (
Δ
A
,
Δ
P
), then the point (
A
=
M
+
Δ
A
,
P
=
N
+
Δ
P
) is a feasible solution to the bilinear matrix inequality (BMI) (7)
l
A
,
P
<
0
.


## 3. SOF Controller Design

In this section, based on a piecewise quadratic Lyapunov function and the new linearization method, we will give new sufficient conditions for solving this problem.Theorem 3.
If there exist points(
F
i
(
k
)
,
P
i
(
k
)
), i
=
1,2
,
…
,
l, such that the following inequalities: (8)
-
P
i
k
+
Δ
P
i
∗
∗
P
j
k
+
Δ
P
j
A
c
i
k
+
P
j
k
B
i
Δ
F
i
C
i
-
P
j
k
+
Δ
P
j
∗
B
i
Δ
F
i
C
i
Δ
P
j
-
2
I
<
0
,
P
i
k
+
Δ
P
i
>
0
,
∀
i
,
j
∈
Ω
,hold for some (
Δ
F
i
,
Δ
P
i
), where A
c
i
(
k
)
=
A
i
+
B
i
F
i
(
k
)
C
i, then the points (
P
i
=
P
i
(
k
)
+
Δ
P
i
,
F
i
=
F
i
(
k
)
+
Δ
F
i
) are feasible solutions to inequalities (5).Proof.
By Schur complement, (5) are equivalent to(9)
-
P
i
A
i
+
B
i
F
i
C
i
T
P
j
P
j
A
i
+
B
i
F
i
C
i
-
P
j
<
0
.Write F
i
=
F
i
(
k
)
+
Δ
F
i, P
i
=
P
i
(
k
)
+
Δ
P
i, and A
c
i
(
k
)
=
A
i
+
B
i
F
i
(
k
)
C
i, where F
i
(
k
) and P
i
(
k
) are fixed matrices. The left side of inequality (9) is expanded around (
F
i
(
k
)
,
P
i
(
k
)
) as(10)
-
P
i
A
i
+
B
i
F
i
C
i
T
P
j
P
j
A
i
+
B
i
F
i
C
i
-
P
j
=
-
P
i
k
+
Δ
P
i
∗
P
j
k
+
Δ
P
j
A
i
+
B
i
F
i
k
+
Δ
F
i
C
i
-
P
j
k
+
Δ
P
j
=
-
P
i
k
+
Δ
P
i
∗
P
j
k
+
Δ
P
j
A
c
i
k
+
P
j
k
B
i
Δ
F
i
C
i
-
P
j
k
+
Δ
P
j
+
l
B
i
Δ
F
i
C
i
0
T
,
0
Δ
P
j
<
0
;that is(11)
l
B
i
Δ
F
i
C
i
0
T
,
0
Δ
P
j
<
P
i
k
+
Δ
P
i
∗
-
P
j
k
+
Δ
P
j
A
c
i
k
-
P
j
k
B
i
Δ
F
i
C
i
P
j
k
+
Δ
P
j
.
Thus, by Lemma2, inequalities (8) hold.
With Theorem3, the nonlinear SOF optimization problem O
P
1
(12)
min
⁡
α
subject  to
P
i
>
0
,
-
P
i
-
α
P
i
A
i
+
B
i
F
i
C
i
T
P
j
P
j
A
i
+
B
i
F
i
C
i
-
P
j
<
0can be replaced by solving the following linear optimization problem O
P
1
′: (13)
min
⁡
α
subject to
P
i
k
+
Δ
P
i
>
0
,
β
P
i
k
Δ
P
i
Δ
P
i
β
P
i
k
>
0
,
-
P
i
k
+
Δ
P
i
-
α
P
i
k
∗
∗
P
j
k
+
Δ
P
j
A
c
i
k
+
P
j
k
B
i
Δ
F
i
C
i
-
P
j
k
+
Δ
P
j
∗
B
i
Δ
F
i
C
i
Δ
P
j
-
2
I
<
0
.Remark 4.
InO
P
1
′, additional inequalities β
P
i
(
k
)
Δ
P
i
Δ
P
i
β
P
i
(
k
)
>
0 are added. This inequalities are the variation of inequalities Δ
P
i
<
β
P
i
(
k
). Due to our new linearization method, the prescribed scalar β should be small [19]. Otherwise, the conservation will increase.


## 4. Extension toH
∞ Control

Consider the discrete-time switched system:(14)
x
k
+
1
=
A
i
x
k
+
B
i
u
k
+
B
w
i
ω
k
,
z
k
=
C
z
i
x
k
+
D
u
i
u
k
,
y
k
=
C
i
x
k
,
for
x
∈
S
i
,
i
=
1,2
,
…
,
l
,where x
(
k
)
∈
R
n, u
(
k
)
∈
R
m, y
(
k
)
∈
R
p, z
(
k
)
∈
R
q, and ω
(
k
)
∈
R
r are the state, the control input, the measured output, the control output, and the disturbance input, respectively; S
i
⊆
R
n, i
=
1,2
,
…
,
l, denotes a partition of the output space into a number of closed polyhedral regions. Let Ω be the set of all possible switches from one region to itself or another region; that is, (15)
Ω
=
i
,
j
:
i
,
j
=
1,2
,
…
,
l
s.t.
x
k
∈
S
i
,
x
k
+
1
∈
S
j
.With the controller (3), the closed-loop system of system (14) becomes(16)
x
k
+
1
=
A
i
+
B
i
F
i
C
i
x
k
+
B
w
i
ω
k
,
z
k
=
C
z
i
+
D
u
i
F
i
C
i
x
k
,
i
=
1,2
,
…
,
l
.In this section, new sufficient conditions for SOF control design for the switched system (14) in the H
∞ framework will be present. Given a scalar γ
>
0, assuming x
(
0
)
=
0, the exogenous signal ω is attenuated by γ if for each integer N
>
0 and for every ω
∈
L
2
(
[
0
,
N
]
,
R
r
)
(17)
∑
k
=
0
N
z
k
2
<
γ
2
∑
k
=
0
N
ω
k
2
.TheH
∞ performance of the closed-loop system (16) proposed by Cuzzola and Morari in [13] is reviewed in the next lemma.Lemma 5 (see [13]).
Consider the switched system (14), if there exists a function V
(
x
)
=
x
T
P
i
x, ∀
x
∈
S
i with P
i
=
P
i
T
>
0 satisfying the following inequality:(18)
V
x
k
+
1
-
V
x
k
<
γ
2
ω
k
2
-
z
k
2
,
∀
k
;then the closed-loop switched system (16) is exponentially stable with H
∞ performance γ.Obviously, inequality (18) is equivalent to the following inequalities:(19)
A
i
+
B
i
F
i
C
i
T
P
j
A
i
+
B
i
F
i
C
i
+
C
z
i
+
D
u
i
F
i
C
i
T
C
z
i
+
D
u
i
F
i
C
i
+
A
i
+
B
i
F
i
C
i
T
P
j
B
w
i
Υ
-
1
B
w
i
T
P
j
A
i
+
B
i
F
i
C
i
-
P
i
<
0
,
∀
i
,
j
∈
Ω
,where Υ
=
γ
2
I
-
B
w
i
T
P
j
B
w
i
>
0.Now, the sufficient conditions to obtain SOF control gains withH
∞ performance are given in the following theorem.Theorem 6.
If there exist points(
F
i
(
k
)
,
P
i
(
k
)
), i
=
1,2
,
…
,
l, such that the LMIs (20)
-
P
i
k
+
Δ
P
i
∗
∗
∗
∗
P
j
k
+
Δ
P
j
A
c
i
k
+
P
j
k
B
i
Δ
F
i
C
i
-
P
j
k
+
Δ
P
j
∗
B
ω
i
T
P
j
k
+
Δ
P
j
A
c
i
k
+
B
ω
i
T
P
j
k
B
i
Δ
F
i
C
i
-
Υ
k
+
B
w
i
T
Δ
P
j
B
w
i
∗
C
z
i
+
D
u
i
F
i
k
+
Δ
F
i
C
i
-
I
B
i
Δ
F
i
C
i
Δ
P
j
Δ
P
j
B
w
i
-
2
I
<
0
,
Υ
k
-
B
w
i
T
Δ
P
j
B
w
i
>
0
,
P
i
k
+
Δ
P
i
>
0
,
∀
i
,
j
∈
Ωhold for some (
Δ
F
i
,
Δ
P
i
), where A
c
i
(
k
)
=
A
i
+
B
i
F
i
(
k
)
C
i and Υ
(
k
)
=
γ
2
I
-
B
w
i
T
P
j
(
k
)
B
w
i
>
0, then the points (
P
i
=
P
i
(
k
)
+
Δ
P
i
,
F
i
=
F
i
(
k
)
+
Δ
F
i
) are feasible solutions to inequalities (19).Proof.
By Schur complement, inequalities (19) are equivalent to(21)
-
P
i
∗
∗
∗
P
j
A
i
+
B
i
F
i
C
i
-
P
j
B
w
i
T
P
j
A
i
+
B
i
F
i
C
i
-
Υ
C
z
i
+
D
u
i
F
i
C
i
-
I
<
0
.
WriteF
i
=
F
i
(
k
)
+
Δ
F
i, P
i
=
P
i
(
k
)
+
Δ
P
i, and A
c
i
(
k
)
=
A
i
+
B
i
F
i
(
k
)
C
i, where F
i
(
k
) and P
i
(
k
) are fixed matrices. The left side of inequality (21) is expanded around (
F
i
(
k
)
,
P
i
(
k
)
) as (22)
-
P
i
k
+
Δ
P
i
∗
∗
∗
P
j
k
+
Δ
P
j
A
i
+
B
i
F
i
k
+
Δ
F
i
C
i
-
P
j
k
+
Δ
P
j
B
w
i
T
P
j
k
+
Δ
P
j
A
i
+
B
i
F
i
k
+
Δ
F
i
C
i
-
Υ
k
+
B
w
i
T
Δ
P
j
B
w
i
C
z
i
+
D
u
i
F
i
k
+
Δ
F
i
C
i
-
I
=
-
P
i
k
+
Δ
P
i
∗
∗
∗
P
j
k
+
Δ
P
j
A
c
i
k
+
P
j
k
B
i
Δ
F
i
C
i
-
P
j
k
+
Δ
P
j
B
w
i
T
P
j
k
+
Δ
P
j
A
c
i
k
+
B
w
i
T
P
j
k
B
i
Δ
F
i
C
i
-
Υ
k
+
B
w
i
T
Δ
P
j
B
w
i
C
z
i
+
D
u
i
F
i
k
+
Δ
F
i
C
i
-
I
+
l
B
i
Δ
F
i
C
i
0
0
0
T
,
0
Δ
P
j
Δ
P
j
B
w
i
0
.
That is, the following inequalities hold:(23)
l
B
i
Δ
F
i
C
i
0
0
0
T
,
0
Δ
P
j
Δ
P
j
B
w
i
0
<
P
i
k
+
Δ
P
i
∗
∗
∗
-
P
j
k
+
Δ
P
j
A
c
i
k
-
P
j
k
B
i
Δ
F
i
C
i
P
j
k
+
Δ
P
j
-
B
w
i
T
P
j
k
+
Δ
P
j
A
c
i
k
-
B
w
i
T
P
j
k
B
i
Δ
F
i
C
i
Υ
k
-
B
w
i
T
Δ
P
j
B
w
i
-
C
z
i
-
D
u
i
F
i
k
+
Δ
F
i
C
i
I
.
Thus, by Lemma2, inequalities (20) hold.
With Theorem6, the nonlinear optimization problem O
P
2 for solving SOF control with H
∞ performance (24)
min
⁡
α
subject  to
P
i
>
0
,
Υ
=
γ
2
I
-
B
w
i
T
P
j
B
w
i
>
0
-
P
i
-
α
P
i
∗
∗
∗
P
j
A
i
+
B
i
F
i
C
i
-
P
j
B
w
i
T
P
j
A
i
+
B
i
F
i
C
i
-
Υ
C
z
i
+
D
u
i
F
i
C
i
-
I
<
0
,
∀
i
,
j
∈
Ωcan be replaced by solving the following linear optimization problem O
P
2
′: (25)
min
⁡
α
subject  to
P
i
k
+
Δ
P
i
>
0
,
Υ
k
-
B
w
i
T
Δ
P
j
B
w
i
>
0
,
β
P
i
k
Δ
P
i
Δ
P
i
β
P
i
k
>
0
,
-
P
i
k
+
Δ
P
i
-
α
P
i
k
∗
∗
∗
∗
P
j
k
+
Δ
P
j
A
c
i
k
+
P
j
k
B
i
Δ
F
i
C
i
-
P
j
k
+
Δ
P
j
∗
B
ω
i
T
P
j
k
+
Δ
P
j
A
c
i
k
+
B
ω
i
T
P
j
k
B
i
Δ
F
i
C
i
-
Υ
k
+
B
w
i
T
Δ
P
j
B
w
i
∗
C
z
i
+
D
u
i
F
i
k
+
Δ
F
i
C
i
-
I
B
i
Δ
F
i
C
i
Δ
P
j
Δ
P
j
B
w
i
-
2
I
<
0
,
∀
i
,
j
∈
Ω
.Based on Theorems3 and 6, an iterative algorithm to solve stabilization and H
∞ control via static output feedback for discrete-time switched systems is established.Algorithm 7.
Consider the following.
Step 1 (initialization step). At initial, we need to obtain initial values of F
i and P
i.
Firstly, letP
i
=
I. Then, solve the optimization problem O
P
1 (or O
P
2 instead for H
∞ case) with respect to F
i and α.
Ifα
<
0, stop; else let F
i
(
0
)
=
F
i, P
i
(
0
)
=
P
i, A
c
i
(
0
)
=
A
i
+
B
i
F
i
(
0
)
C
i, and α
(
0
)
=
α, and set k
=
1.
Step 2 (small perturbation step). Set β
=
β
0, where β
0
>
0 is a prescribed small value. Solve LMI optimization problem O
P
1
′ (or O
P
2
′ instead for H
∞ case) with respect to Δ
F
i, Δ
P
i, and α.
Step 3 (update step). Let F
i
(
k
)
=
F
i
(
k
-
1
)
+
Δ
F
i, P
i
(
k
)
=
P
i
(
k
-
1
)
+
Δ
P
i, α
(
k
)
=
α, and A
c
i
(
k
)
=
A
c
i
(
k
-
1
)
+
B
i
Δ
F
i
C
i. For fixed F
i
(
k
), compute new P
i
(
k
) by solving O
P
1 (or O
P
2 instead for H
∞ case), and then compute new F
i
(
k
) and α
(
k
) by solving O
P
1 (or O
P
2 instead for H
∞ case).
Ifα
(
k
)
<
0, stop; else if the relative improvement in α is more than a desired accuracy, set k
=
k
+
1, and go to Step 2. Else, set k
=
1, and let F
i
(
0
)
=
F
i
(
k
), P
i
(
0
)
=
P
i
(
k
), A
c
i
(
0
)
=
A
c
i
(
k
), and α
(
0
)
=
α
(
k
).
Step 4 (wide perturbation step). Set β
=
β
×
2. Solve LMI optimization problem O
P
1
′ (or O
P
2
′ instead for H
∞ case) with respect to Δ
F
i, Δ
P
i, and α.
Step 5 (update step). Let F
i
(
k
)
=
F
i
(
k
-
1
)
+
Δ
F
i, P
i
(
k
)
=
P
i
(
k
-
1
)
+
Δ
P
i, α
(
k
)
=
α, and A
c
i
(
k
)
=
A
c
i
(
k
-
1
)
+
B
i
Δ
F
i
C
i. For fixed F
i
(
k
), compute new P
i
(
k
) by solving O
P
1 (or O
P
2 instead for H
∞ case), and then compute new F
i
(
k
) and α
(
k
) by solving O
P
1 (or O
P
2 instead for H
∞ case).
Ifα
(
k
)
<
0, stop; else if the relative improvement in α is more than a desired accuracy, set k
=
1, and let F
i
(
0
)
=
F
i
(
k
), P
i
(
0
)
=
P
i
(
k
), A
c
i
(
0
)
=
A
c
i
(
k
), and α
(
0
)
=
α
(
k
), and go to Step 2. Else if the relative improvement in α is inferior to the desired accuracy and k
<
k
0, where k
0
>
0 is a prescribed integer, set k
=
k
+
1, and go to Step 4. Else, stop.Remark 8.
The wide perturbation step is a crucial step in improved path-following method. The purpose of this step is to broaden the search scope during each iteration so that the algorithm has the opportunity to escape form the local optimum. However, the enlarged search scope may cause nonconvergence. So the iteration number of wide perturbation step should not be too large. As long as the objective functionα has been improved significantly, the wide perturbation step will be replaced by a small perturbation step immediately.


## 5. Numerical Examples

In this section, two examples are given to show the effectiveness of our method. Example1 is with respect to the SOF control problem for switched systems. Example 2 is concerning the H
∞ controller design problem for switched systems.Example 1.
Consider system (1) with the following parameters: (26)
A
1
=
1
0.3
2
1
0
1
0.3
0.6
0.6
,
A
2
=
-
0.5871
-
0.8441
-
0.0092
-
0.6865
-
0.5090
-
0.8561
0.0974
0.4523
-
0.2280
,
A
3
=
0.1089
0.2458
-
0.9035
0.3998
-
0.9213
-
0.4161
0.6745
-
0.5750
0.7138
,
B
1
=
1
0
0
1
1
0
,
B
2
=
0.1930
-
0.4204
-
0.7359
0.0346
0.5073
-
0.9077
,
B
3
=
-
0.4164
0.0244
0.8297
-
0.4366
-
0.0900
-
0.8416
,
C
1
=
1
1
0
,
C
2
=
1
0
1
,
C
3
=
0
1
1
.It cannot be stabilized by the method in [12, 20]. However, using our method, set β
0
=
0.2; after iterations, output feedback stabilizing controller matrices are computed to be (27)
F
1
=
-
0.7454
-
0.8806
,
F
2
=
-
0.8899
-
0.2598
,
F
3
=
0.5736
0.4941
,
P
1
=
11359.9012
-
13835.4590
3078.1446
-
13835.4590
45253.4272
-
20668.2151
3078.1446
-
20668.2151
36198.7545
,
P
2
=
7437.9005
-
6354.0674
-
141.5298
-
6354.0674
34828.7955
-
8170.2774
-
141.5298
-
8170.2774
16785.3283
,
P
3
=
12867.1197
-
18832.8739
5382.2570
-
18832.8739
43961.3298
-
14881.7520
5382.2570
-
14881.7520
23348.2946
.Example 2.
Consider a system with the following parameters:(28)
A
1
=
-
0.5871
-
0.8441
-
0.0092
-
0.6865
-
0.5090
-
0.8561
0.0974
0.4523
-
0.2280
,
A
2
=
0.1089
0.2458
-
0.9035
0.3998
-
0.9213
-
0.4161
0.6745
-
0.5750
0.7138
,
B
1
=
0.1930
-
0.4204
-
0.7359
0.0346
0.5073
-
0.9077
,
B
2
=
-
0.4164
0.0244
0.8297
-
0.4366
-
0.0900
-
0.8416
,
C
1
=
1
0
1
,
C
2
=
0
1
1
,
B
w
1
=
B
w
2
=
0
1
0
,
C
z
1
=
C
z
2
=
1
0
0
0
1
0
0
0
1
,
D
u
1
=
D
u
2
=
0
0
0
1
0
1
.Using our method, setβ
0
=
0.2; after iterations, output feedback control matrices are computed to be (29)
F
1
=
-
0.9811
-
0.6662
,
F
2
=
0.4284
-
0.2097
.In this case, (30)
P
1
=
5.3254
-
2.9578
1.2843
-
2.9578
12.6330
-
2.3378
1.2843
-
2.3378
2.8444
,
P
2
=
6.0132
-
3.0439
3.3919
-
3.0439
6.3742
-
3.3469
3.3919
-
3.3469
9.2144and the closed-loop system has the H
∞ performance γ
=
4.7328. The result is better than the solution solved by the method in [12] which gives γ
=
5.6853, which is inferior compared to our result γ
=
4.7328 and (31)
F
1
=
-
1.0832
-
0.5259
,
F
2
=
0.3563
-
0.1241
.


## 6. Conclusion

This paper studies the problems of static output feedback control andH
∞ controller synthesis for discrete-time switched systems. Based on piecewise quadratic Lyapunov functions and a new linearization method, new sufficient conditions for system stability and H
∞ controller design are obtained. Then, an improved path-following algorithm is built to solve the problems. Finally, the merits and effectiveness of the proposed method are shown by two numerical examples. Compared to the existing methods, the proposed method is less conservative.Important future research work will be applying the results to some real-world systems. How to reduce the design conservatism is an important research topic that deserves further investigation.



---

*Source: 869012-2015-10-11.xml*