# Parameterized Regulator Synthesis for Bimodal Linear Systems Based on Bilinear

Authors: Zhizheng Wu; Foued Ben Amara
Journal: Mathematical Problems in Engineering
 (2008)

---

## Abstract

A regulator design method is presented for switched bimodal linear systems, where it is desired to reject known disturbance signals and/or track known reference inputs. The switching in the bimodal system is defined by a switching surface. The regulator design approach consists of three steps. The first step is based on constructing a switched observer-based state feedback central controller for the switched linear system. The second step involves augmenting the switched central controller with additional dynamics to construct a parameterized set of switched controllers. In the third step, two sufficient regulation conditions are derived for the resulting switched closed loop system. The regulation conditions present guidelines for the selection of the additional dynamics used to parameterize the switched controllers to yield the desired regulator. A regulator synthesis approach is proposed based on solving properly formulated bilinear matrix inequalities. Finally, a numerical example is presented to illustrate the performance of the proposed regulator.


---

## Body

## 1. Introduction

Recently, switched control systems have attracted much attention in
the control community since they present challenging problems of practical
importance. Significant progress has been made in this area, most notably in
the stability analysis for such systems [1–7]. Numerous other results
have been published and are presented in [8–10] and the references therein.
In practice, in addition to stability requirements, there is a need to find
controllers that would achieve regulation against known reference or
disturbance signals. For example, in hard disk drives, maintaining a constant
small distance between the read/write head and the disk surface is an important
target that would allow greater data storage densities on the hard disk. However,
the close proximity of the read/write head to the disk surface leads to
intermittent contact between the two and results in a switched system regulation
problem [11, 12]. Solutions to the output regulation problem for the
case of
linear systems have been presented in [13–15] and for the case of
nonlinear systems in
[16, 17]. However, these results cannot be applied to switched systems
since the vector fields of switched systems are discontinuous and nonsmooth. The
work of Devasia et al. [18] and
Sakurama and  Sugie [19] are closely related to this problem.
Devasia et al. [18] studied the exact output tracking problem for linear switched
systems and presented necessary and sufficient conditions for the existence of a
regulator for exact output tracking with parameter jumps. The derived solution
was based on the assumption that the switching times are known a priori.
Therefore, the necessary and sufficient conditions were derived only for the
elimination of switching induced output transients and cannot be applied to systems
where switching is signal driven or where the switching times cannot be set a
priori. Sakurama and Sugie [19] recently discussed the trajectory tracking
problem for bimodal switched systems. The reference trajectory for the switched
system is assumed known and the switching times are assumed to be measurable. An
error variable and an error system are introduced, based on which the tracking
controller was designed using a Lyapunov-like function. However, conditions for
the existence of the desired regulator are sometimes very difficult to check.In this
paper, a controller design approach is proposed for single-input single-output switched
bimodal systems, where it is desired to reject known disturbance signals and
track known reference inputs simultaneously. Switching in the bimodal system is defined by a switching surface. A
regulator synthesis method based on solving bilinear matrix inequalities (BMIs)
is presented. The proposed regulator design approach
consists of three steps. In the first step, a switched observer-based
state-feedback central controller is constructed for the switched linear
system. The second step involves augmenting the switched central controller with
additional dynamics to construct a parameterized set of switched controllers. Stability
analysis of the resulting switched closed loop system is then presented. In the
third step, two sufficient regulation
conditions are derived for the switched closed loop system. The first
sufficient condition is derived based on the input-output stability property of
the switched closed loop system. The second sufficient condition for regulation is derived by transforming the
forced switched closed loop system into an unforced impulsive switched system.
As such, the regulation problem is transformed into a stability analysis
problem for the impulsive switched system. Based on the parameterized
controller structure and the derived regulation conditions, proper BMIs are formulated
and a regulator synthesis method is proposed. The main advantage of the
proposed regulator synthesis approach is that it offers a numerical procedure
that can practically be implemented and used to develop the desired switched
regulator.The
rest of the paper is organized as follows. In Section2, the general regulation
problem for switched bimodal linear systems is presented. In Section 3, the
construction of a parameterized set of switched controllers for the switched
system is discussed and the stability properties of the resulting closed loop
switched system are analyzed. Regulation conditions for the switched system are
presented in Section 4 and the regulator synthesis method for switched systems
is proposed in Section 5. The controller design method is illustrated in
Section
6 using a numerical example, followed by the conclusion in Section 7.In the
following,λ
max
(
⋅
) and λ
min
(
⋅
)denote the maximum and minimum eigenvalues of
a matrix, respectively. The symbol I represents the identity matrix. For a given 1
×
Nmatrix X,
X
→ denotes the N
×
1vector with the same entries as those of X
T.


## 2. Regulation Problem for Switched Bimodal Systems

This paper considers switched
bimodal systems subject to external inputs representing disturbance and/or
reference signals. Both the system dynamics as well as the external input
signal are assumed to switch according to a switching law defined by a
switching surface. These types of switched systems are motivated by practical
applications, such as the flying height regulation problem for the read/write head
in hard disk drives [11, 12].Consider
the switched system given by the following state space representation:(2.1)
Σ
r
:
{
x
˙
=
A
r
x
+
B
r
u
+
D
r
x
w
r
,
x
(
0
)
=
x
0
,
w
˙
r
=
H
r
w
r
,
w
r
(
0
)
=
w
r
0
,
y
=
C
r
y
x
+
D
r
y
w
r
,
e
=
C
e
x
+
D
r
e
w
r
,
r
=
{
1
if
e
=
C
e
x
+
D
r
e
w
r
≤
δ
,
2
if
e
=
C
e
x
+
D
r
e
w
r
>
δ
, where x
∈
ℝ
n is the state vector, u
∈
ℝ is the control input, y
∈
ℝ is the measurement signal to be fed to the
controller, e
∈
ℝ is the performance variable to be regulated
and is assumed to be measurable, w
r
∈
ℝ
h is the state vector of the exogenous systems
generating the signal w
r, H
r
∈
ℝ
h
×
h have simple eigenvalues on the imaginary axis, r
∈
{
1
,
2
} is the index of the system Σ
r under consideration at time t,
and δ is a constant satisfying |
δ
|
>
0.
The switching between the systems Σ
1 and Σ
2 is performed according to the value of the
performance variable e,
and is determined based on the location of x with respect to a switching surface S given by (2.2)
S
=
{
x
∈
ℝ
n
:
e
=
C
e
x
+
D
r
e
w
r
=
δ
}
.The switching surface S is not fixed but changes with time given that
the term D
r
e
w
r is in general a time-varying term. The switching between the two modes
takes place as follows. If r
=
1 and e becomes strictly greater than δ,
then the mode switches to r
=
2;
and if r
=
2 and e becomes less than or equal to δ,
then the mode switches to r
=
1.
In the following, it is assumed that for any given t
≥
0,
the system must operate in only one of the two modes corresponding to r
∈
{
1
,
2
}.
This assumption is motivated by physical considerations in some applications of
interest, such as the example presented at the end of the paper or the system
treated in [11, 12]. For the switched system
(2.1), it is desired
to construct an output feedback controller to regulate the performance variable e of the switched system against the external
input signal w
r.
Given the switching nature of the plant, the output feedback controller is also
chosen to be a switching feedback controller ℂ
r, r
∈
{
1
,
2
},
with a state space representation given by(2.3)
ℂ
r
:
{
x
˙
c
=
A
r
,
c
x
c
+
B
r
,
c
y
,
u
=
C
r
,
c
x
c
, where x
c
∈
ℝ
n
c,
and where the switching among controllers is to obey the same rule given in (2.1) for switching between the two plant
models. Therefore, the resulting closed loop system is given by(2.4)
Σ
r
cl
:
{
[
x
˙
x
˙
c
]
=
[
A
r
B
r
C
r
,
c
B
r
,
c
C
r
y
A
r
,
c
]
[
x
x
c
]
+
[
D
r
x
B
r
,
c
D
r
y
]
w
r
,
e
=
[
C
e
0
]
[
x
x
c
]
+
D
r
e
w
r
,
r
=
{
1
if
e
=
C
e
x
+
D
r
e
w
r
≤
δ
,
2
if
e
=
C
e
x
+
D
r
e
w
r
>
δ
. The output feedback regulation
problem for the switched system (2.1) can then be stated as follows.Output-Feedback Regulation Problem
Given the switched system (2.1), find a switched output feedback
controller of the form (2.3)
such that the resulting closed loop system satisfies the following conditions.
(C1)
Internal stability. With w
r
(
0
)
=
0,
the equilibrium point [
x
T
x
c
T
]
T
=
0 of the unforced switched closed loop system is
exponentially stable.
(C2)
Output regulation. For
all [
x
T
(0)
x
c
T
(
0
)
]
T
∈
ℝ
n
×
ℝ
n
c and w
r
(
0
)
∈
ℝ
h,
the response e
(
t
) of the closed loop system involving the
switched system (2.1) and the switched controller (2.3) satisfies (2.5)
lim
⁡
t
→
∞
e
(
t
)
=
0.             In the following
section, a
framework within which regulation conditions will be derived is presented.


## 3. Parameterization of a Set of Controllers

The
controller design approach presented in this paper relies on the construction
of a parameterized set of output feedback controllers for the switched system (2.1). In this
section, the construction of such
a set is first discussed, followed by an analysis of the stability properties
of the resulting switched closed loop system. The construction of a
parameterized set of switched controllers involves two steps. The first step consists
of designing a central controller in the form of an observer-based state-feedback
controller. The second step involves augmenting the central controller with
additional dynamics to construct a parameterized set of controllers.


### 3.1. Observer-Based State-Feedback Controller

Consider
the following observer-based state-feedback controller for the switched system (2.1) (3.1)
ℂ
r
o
:
{
x
^
˙
=
A
r
x
^
+
B
r
u
+
L
r
(
y
^
−
y
)
,

x
^
(
0
)
=
x
^
0
,
u
=
K
r
x
^
, where x
^
∈
ℝ
n is the estimate of the plant state vector x and y
^
=
C
r
y
x
^ is an estimate of the plant output y.
The mode r
∈
{
1
,
2
} is determined according to the rule given in
(2.1). Moreover,
it is assumed that there are no impulsive changes in the controller states at
the switching times.


### 3.2. Parameterized Output-Feedback Controller

The
construction of a parameterized set of controllers for the switched system is
based on considering, for eachr
∈
{
1
,
2
},
a linear fractional transformation involving a fixed system 𝕁
rand a proper system ℚ
r as shown in Figure 1. The
proposed controllers are similar to the parameterized stabilizing controllers
for linear systems
[20, 21], but where no stability assumptions are placed on the system ℚ
r.
The state space representation of the system 𝕁
r is given by (3.2)
J
r
:
{
x
^
˙
=
(
A
r
+
B
r
K
r
+
L
r
C
r
y
)
x
^
−
L
r
y
+
[
B
r
(
L
r
D
r
y
+
D
r
x
)
]
y
Q
,

x
^
(
0
)
=
x
^
0
,
u
=
K
r
x
^
+
[
1
0
1
×
q
]
y
Q
,
y
^
−
y
=
C
r
y
x
^
−
y
, and the
system ℚ
r is given by(3.3)
ℚ
r
:
{
x
˙
Q
=
A
Q
r
x
Q
+
B
Q
r
(
y
^
−
y
)
,

x
Q
(
0
)
=
x
Q
0
,
y
Q
=
[
C
Q
r
I
q
×
q
]
x
Q
, where x
Q
∈
ℝ
q and y
Q
∈
ℝ
q
+
1.
In particular, throughout the rest of the paper, the system ℚ
r is such that the matrices A
Q
r, B
Q
r,
and C
Q
r also change according to r.Figure 1
Closed loop system with a parameterized controller.Remark 3.1.
The state space representation
of the system𝕁
r given in (3.2) differs from that traditionally used
in the construction of parameterized sets of stabilizing controllers for linear
systems in that the state equation in (3.2) contains the additional term (
L
r
D
r
y
+
D
r
x
)
x
Q.
The presence of this term makes it possible to derive the sufficient conditions
for regulation presented in Theorem 4.2.By
combining the systems𝕁
r and ℚ
r in (3.2) and (3.3), the state-space representation of the
regulator is then given by (2.3),
where(3.4)
x
c
=
[
x
^
x
Q
]
,


A
r
,
c
=
[
A
r
+
B
r
K
r
+
L
r
C
r
y
B
r
C
Q
r
+
L
r
D
r
y
+
D
r
x
B
Q
r
C
r
y
A
Q
r
]
,
B
r
,
c
=
[
−
L
r
−
B
Q
r
]
,


C
r
,
c
=
[
K
r
C
Q
r
]
,


D
r
,
c
=
0. Let x
=
x
^
−
x denote the state estimation error. It follows
that the resulting closed loop system involving the plant (2.1) and the regulator given by (2.3) and
(3.4) can be written as follows:(3.5)
Σ
r
cl
:
{
[
x
˙
x
˙
x
˙
Q
]
=
[
A
r
+
B
r
K
r
B
r
K
r
B
r
C
Q
r
0
A
r
+
L
r
C
r
y
D
r
x
+
L
r
D
r
y
0
B
Q
r
C
r
y
A
Q
r
]
[
x
x
x
Q
]
+
[
D
r
x
−
D
r
x
−
L
r
D
r
y
−
B
Q
r
D
r
y
]
w
r
,
e
=
[
C
e
0
0
]
[
x
x
x
Q
]
+
D
r
e
w
r
. Let χ
=
[
x
T
x
T
x
Q
T
]
1
×
N
T denote the state vector of the resulting
closed loop system with N
=
2
n
+
q,
and let(3.6)
A
^
r
=
[
A
r
+
B
r
K
r
B
r
K
r
B
r
C
Q
r
0
A
r
+
L
r
C
r
y
D
r
x
+
L
r
D
r
y
0
B
Q
r
C
r
y
A
Q
r
]
,
E
r
=
[
(
D
r
x
)
T
−
(
D
r
x
+
L
r
D
r
y
)
T
−
(
B
Q
r
D
r
y
)
T
]
T
,


C
e
x
=
[
C
e
0
0
]
. The
resulting parameterized switched closed loop system dynamics can then be
expressed as (3.7)
Σ
r
cl
:
{
χ
˙
=
A
^
r
χ
+
E
r
w
r
,
e
=
C
e
x
χ
+
D
r
e
w
r
,
r
=
{
1
if
e
=
C
e
x
χ
+
D
r
e
w
r
≤
δ
,
2
if
e
=
C
e
x
χ
+
D
r
e
w
r
>
δ
. In the
following, both the internal stability as well as the input-output stability
properties of the system (3.7)
are analyzed.


### 3.3. Stability of the Parameterized Switched Closed Loop System

In this section, it is desired
to study the stability properties of the system (3.7). The stability analysis involves two
steps, namely, internal stability analysis and input-output stability analysis.
In the first step, the internal stability of the closed loop system is studied
by considering the system (3.7)
in the absence of the signal w
r,
and studying the stability properties of the origin for the resulting unforced
switched system. The second step builds on the internal stability results and
presents input-output stability results for the system (3.7).


#### 3.3.1. Internal Stability

Consider first the system (3.7) in the absence of the signal w
r. The state equation for the resulting system is given by(3.8)
χ
˙
=
A
^
r
χ
.Note that, there are no
impulsive changes in the state variables of the above switched system at the
switching times. The internal stability of the system (3.8) is then given by the following well
known result.Lemma 3.2 (see [22]).
If there
exists a matrixP
=
P
T
>
0 and a constant α such that the following inequalities hold (3.9)
A
^
r
T
P
+
P
A
^
r
+
2
α
P
<
0
,

α
>
0,
r
∈
{
1
,
2
}
, then the origin is an exponentially stable
equilibrium point for the switched system (3.8) with arbitrary switching, and(3.10)
‖
χ
(
t
)
‖
<
c
e
−
α
t
‖
χ
(
0
)
‖
,where
c
=
λ
max
(
P
)
/
λ
min
(
P
).Remark 3.3.
The matrix inequality (3.9) can be solved using numerical
algorithms. Here, we search for a common Lyapunov function for the switched
systems (3.8) to guarantee the stability of the
origin for the system (3.8)
under arbitrary switching, since the switching surface S
=
{
x
∈
ℝ
n
:
e
=
C
e
x
+
D
r
e
w
r
=
δ
} is not a fixed function of the state vector x and changes with D
r
e
w
r.


#### 3.3.2. Input-Output Stability

In this
section, we will consider the input-output stability properties of the system (3.7) with the external input signal w
r.
Let w
m
=
max
r
∈
{
1
,
2
}
;
t
≥
0
∥
E
r
w
r
(
t
)
∥.
Then we have the following stability result.Lemma 3.4 (see [9]).
Assume
the origin for the switched system (3.8) is an exponentially stable
equilibrium point. Then the state vector in (3.7) is bounded, and the states will
ultimately evolve inside the bounded set B
β given by(3.11)
B
β
=
{
χ
∈
ℝ
N
:
‖
χ
‖
≤
β
}
, where β
=
c
w
m
/
α.
The input-output stability of
the closed loop system follows immediately from the above result.


## 3.1. Observer-Based State-Feedback Controller

Consider
the following observer-based state-feedback controller for the switched system (2.1) (3.1)
ℂ
r
o
:
{
x
^
˙
=
A
r
x
^
+
B
r
u
+
L
r
(
y
^
−
y
)
,

x
^
(
0
)
=
x
^
0
,
u
=
K
r
x
^
, where x
^
∈
ℝ
n is the estimate of the plant state vector x and y
^
=
C
r
y
x
^ is an estimate of the plant output y.
The mode r
∈
{
1
,
2
} is determined according to the rule given in
(2.1). Moreover,
it is assumed that there are no impulsive changes in the controller states at
the switching times.


## 3.2. Parameterized Output-Feedback Controller

The
construction of a parameterized set of controllers for the switched system is
based on considering, for eachr
∈
{
1
,
2
},
a linear fractional transformation involving a fixed system 𝕁
rand a proper system ℚ
r as shown in Figure 1. The
proposed controllers are similar to the parameterized stabilizing controllers
for linear systems
[20, 21], but where no stability assumptions are placed on the system ℚ
r.
The state space representation of the system 𝕁
r is given by (3.2)
J
r
:
{
x
^
˙
=
(
A
r
+
B
r
K
r
+
L
r
C
r
y
)
x
^
−
L
r
y
+
[
B
r
(
L
r
D
r
y
+
D
r
x
)
]
y
Q
,

x
^
(
0
)
=
x
^
0
,
u
=
K
r
x
^
+
[
1
0
1
×
q
]
y
Q
,
y
^
−
y
=
C
r
y
x
^
−
y
, and the
system ℚ
r is given by(3.3)
ℚ
r
:
{
x
˙
Q
=
A
Q
r
x
Q
+
B
Q
r
(
y
^
−
y
)
,

x
Q
(
0
)
=
x
Q
0
,
y
Q
=
[
C
Q
r
I
q
×
q
]
x
Q
, where x
Q
∈
ℝ
q and y
Q
∈
ℝ
q
+
1.
In particular, throughout the rest of the paper, the system ℚ
r is such that the matrices A
Q
r, B
Q
r,
and C
Q
r also change according to r.Figure 1
Closed loop system with a parameterized controller.Remark 3.1.
The state space representation
of the system𝕁
r given in (3.2) differs from that traditionally used
in the construction of parameterized sets of stabilizing controllers for linear
systems in that the state equation in (3.2) contains the additional term (
L
r
D
r
y
+
D
r
x
)
x
Q.
The presence of this term makes it possible to derive the sufficient conditions
for regulation presented in Theorem 4.2.By
combining the systems𝕁
r and ℚ
r in (3.2) and (3.3), the state-space representation of the
regulator is then given by (2.3),
where(3.4)
x
c
=
[
x
^
x
Q
]
,


A
r
,
c
=
[
A
r
+
B
r
K
r
+
L
r
C
r
y
B
r
C
Q
r
+
L
r
D
r
y
+
D
r
x
B
Q
r
C
r
y
A
Q
r
]
,
B
r
,
c
=
[
−
L
r
−
B
Q
r
]
,


C
r
,
c
=
[
K
r
C
Q
r
]
,


D
r
,
c
=
0. Let x
=
x
^
−
x denote the state estimation error. It follows
that the resulting closed loop system involving the plant (2.1) and the regulator given by (2.3) and
(3.4) can be written as follows:(3.5)
Σ
r
cl
:
{
[
x
˙
x
˙
x
˙
Q
]
=
[
A
r
+
B
r
K
r
B
r
K
r
B
r
C
Q
r
0
A
r
+
L
r
C
r
y
D
r
x
+
L
r
D
r
y
0
B
Q
r
C
r
y
A
Q
r
]
[
x
x
x
Q
]
+
[
D
r
x
−
D
r
x
−
L
r
D
r
y
−
B
Q
r
D
r
y
]
w
r
,
e
=
[
C
e
0
0
]
[
x
x
x
Q
]
+
D
r
e
w
r
. Let χ
=
[
x
T
x
T
x
Q
T
]
1
×
N
T denote the state vector of the resulting
closed loop system with N
=
2
n
+
q,
and let(3.6)
A
^
r
=
[
A
r
+
B
r
K
r
B
r
K
r
B
r
C
Q
r
0
A
r
+
L
r
C
r
y
D
r
x
+
L
r
D
r
y
0
B
Q
r
C
r
y
A
Q
r
]
,
E
r
=
[
(
D
r
x
)
T
−
(
D
r
x
+
L
r
D
r
y
)
T
−
(
B
Q
r
D
r
y
)
T
]
T
,


C
e
x
=
[
C
e
0
0
]
. The
resulting parameterized switched closed loop system dynamics can then be
expressed as (3.7)
Σ
r
cl
:
{
χ
˙
=
A
^
r
χ
+
E
r
w
r
,
e
=
C
e
x
χ
+
D
r
e
w
r
,
r
=
{
1
if
e
=
C
e
x
χ
+
D
r
e
w
r
≤
δ
,
2
if
e
=
C
e
x
χ
+
D
r
e
w
r
>
δ
. In the
following, both the internal stability as well as the input-output stability
properties of the system (3.7)
are analyzed.


## 3.3. Stability of the Parameterized Switched Closed Loop System

In this section, it is desired
to study the stability properties of the system (3.7). The stability analysis involves two
steps, namely, internal stability analysis and input-output stability analysis.
In the first step, the internal stability of the closed loop system is studied
by considering the system (3.7)
in the absence of the signal w
r,
and studying the stability properties of the origin for the resulting unforced
switched system. The second step builds on the internal stability results and
presents input-output stability results for the system (3.7).


### 3.3.1. Internal Stability

Consider first the system (3.7) in the absence of the signal w
r. The state equation for the resulting system is given by(3.8)
χ
˙
=
A
^
r
χ
.Note that, there are no
impulsive changes in the state variables of the above switched system at the
switching times. The internal stability of the system (3.8) is then given by the following well
known result.Lemma 3.2 (see [22]).
If there
exists a matrixP
=
P
T
>
0 and a constant α such that the following inequalities hold (3.9)
A
^
r
T
P
+
P
A
^
r
+
2
α
P
<
0
,

α
>
0,
r
∈
{
1
,
2
}
, then the origin is an exponentially stable
equilibrium point for the switched system (3.8) with arbitrary switching, and(3.10)
‖
χ
(
t
)
‖
<
c
e
−
α
t
‖
χ
(
0
)
‖
,where
c
=
λ
max
(
P
)
/
λ
min
(
P
).Remark 3.3.
The matrix inequality (3.9) can be solved using numerical
algorithms. Here, we search for a common Lyapunov function for the switched
systems (3.8) to guarantee the stability of the
origin for the system (3.8)
under arbitrary switching, since the switching surface S
=
{
x
∈
ℝ
n
:
e
=
C
e
x
+
D
r
e
w
r
=
δ
} is not a fixed function of the state vector x and changes with D
r
e
w
r.


### 3.3.2. Input-Output Stability

In this
section, we will consider the input-output stability properties of the system (3.7) with the external input signal w
r.
Let w
m
=
max
r
∈
{
1
,
2
}
;
t
≥
0
∥
E
r
w
r
(
t
)
∥.
Then we have the following stability result.Lemma 3.4 (see [9]).
Assume
the origin for the switched system (3.8) is an exponentially stable
equilibrium point. Then the state vector in (3.7) is bounded, and the states will
ultimately evolve inside the bounded set B
β given by(3.11)
B
β
=
{
χ
∈
ℝ
N
:
‖
χ
‖
≤
β
}
, where β
=
c
w
m
/
α.
The input-output stability of
the closed loop system follows immediately from the above result.


## 3.3.1. Internal Stability

Consider first the system (3.7) in the absence of the signal w
r. The state equation for the resulting system is given by(3.8)
χ
˙
=
A
^
r
χ
.Note that, there are no
impulsive changes in the state variables of the above switched system at the
switching times. The internal stability of the system (3.8) is then given by the following well
known result.Lemma 3.2 (see [22]).
If there
exists a matrixP
=
P
T
>
0 and a constant α such that the following inequalities hold (3.9)
A
^
r
T
P
+
P
A
^
r
+
2
α
P
<
0
,

α
>
0,
r
∈
{
1
,
2
}
, then the origin is an exponentially stable
equilibrium point for the switched system (3.8) with arbitrary switching, and(3.10)
‖
χ
(
t
)
‖
<
c
e
−
α
t
‖
χ
(
0
)
‖
,where
c
=
λ
max
(
P
)
/
λ
min
(
P
).Remark 3.3.
The matrix inequality (3.9) can be solved using numerical
algorithms. Here, we search for a common Lyapunov function for the switched
systems (3.8) to guarantee the stability of the
origin for the system (3.8)
under arbitrary switching, since the switching surface S
=
{
x
∈
ℝ
n
:
e
=
C
e
x
+
D
r
e
w
r
=
δ
} is not a fixed function of the state vector x and changes with D
r
e
w
r.


## 3.3.2. Input-Output Stability

In this
section, we will consider the input-output stability properties of the system (3.7) with the external input signal w
r.
Let w
m
=
max
r
∈
{
1
,
2
}
;
t
≥
0
∥
E
r
w
r
(
t
)
∥.
Then we have the following stability result.Lemma 3.4 (see [9]).
Assume
the origin for the switched system (3.8) is an exponentially stable
equilibrium point. Then the state vector in (3.7) is bounded, and the states will
ultimately evolve inside the bounded set B
β given by(3.11)
B
β
=
{
χ
∈
ℝ
N
:
‖
χ
‖
≤
β
}
, where β
=
c
w
m
/
α.
The input-output stability of
the closed loop system follows immediately from the above result.


## 4. Regulation Conditions for the Switched System

The purpose of introducing the
parameterized controllers is to find, for eachr
∈
{
1
,
2
},
appropriate A
Q
r, B
Q
r,
and C
Q
r in (3.3) to solve the output-feedback
regulation problem for the switched system (3.7). The solution to the regulation
problem for the switched system (3.7) is presented in two steps. In the
first step, regulation conditions for each of the individual closed loop systems Σ
r
cl,  r
∈
{
1
,
2
},
are presented. In the second step, two sufficient conditions for regulation in
the switched closed loop systems Σ
r
cl are derived. To derive the sufficient
condition for regulation, a coordinate transformation is defined first,
allowing the forced switched closed loop system to be transformed into an
unforced impulsive switched system. Hence, the regulation problem is
transformed into a stability analysis problem for the origin of the resulting impulsive
switched system. Conditions for achieving asymptotic stability in the new
impulsive switched system are then presented, which is equivalent to achieving
regulation in the original switched closed loop system.


### 4.1. Regulation Conditions forΣ
1
cland Σ
2
cl

LetA
¯
r
=
[
A
r
B
r
C
r
,
c
B
r
,
c
C
r
y
A
r
,
c
] and E
¯
r
=
[
D
r
x
B
r
,
c
D
r
y
] in (2.4). For each of the systems Σ
r, r
∈
{
1
,
2
},
and using a controller of the form (2.3), regulation conditions are given by
the following lemma.Lemma 4.1 (see [15]).
For eachr
∈
{
1
,
2
},
consider the system Σ
r in (2.1) and a controller ℂ
r in (2.3), and assume that A
¯
r is a stability matrix. Then, for each r
∈
{
1
,
2
}, Σ
r
cl in (2.4) achieves regulation if and only if
there exists a matrix Π
r that solves the linear matrix equations:(4.1)
Π
r
H
r
=
A
¯
r
Π
r
+
E
¯
r
,
0
=
C
e
x
Π
r
+
D
r
e
.
Similarly,
using the parameterized feedback regulator ℂ
r given in (2.3) and (3.4), regulation conditions for the
switched closed loop system (3.7)
can be derived and are presented in the following theorem.Theorem 4.2.
Assume
that for eachr
∈
{
1
,
2
},
the closed loop system Σ
r
cl given in (3.7) is exponentially stable. Then each of
the systems Σ
1
cl and Σ
2
cl achieves regulation only if,
for each r
∈
{
1
,
2
},
there exists a pair of matrices (
Ω
r
,
Ψ
r
) which satisfy the following equations:(4.2)
Ω
r
H
r
=
A
r
Ω
r
+
B
r
Ψ
r
+
D
r
x
,
0
=
C
e
Ω
r
+
D
r
e
. Furthermore,
if the linear matrix equations in (4.2) admit a solution (
Ω
r
,
Ψ
r
),
and the matrices A
Q
r and C
Q
r in (3.3) are taken to be of the form(4.3)
A
Q
r
=
H
r
+
B
Q
r
D
r
y
,
C
Q
r
=
Ψ
r
−
K
r
Ω
r
, then,
for each r
∈
{
1
,
2
}, Σ
r
cl in (3.7) achieves regulation.Proof.
Equation
(4.1) can be rewritten as(4.4)
Π
r
H
r
=
[
A
r
B
r
C
r
,
c
B
r
,
c
C
r
y
A
r
,
c
]
Π
r
+
[
D
r
x
B
r
,
c
D
r
y
]
,
0
=
[
C
e
0
]
Π
r
+
D
r
e
. Partition Π
r as Π
r
=
[
Ω
r
Φ
r
].
Then (4.1) is the same as(4.5)
Ω
r
H
r
=
A
r
Ω
r
+
B
r
C
r
,
c
Φ
r
+
D
r
x
,
Φ
r
H
r
=
A
r
,
c
Φ
r
+
B
r
,
c
(
C
r
y
Ω
r
+
D
r
y
)
,
0
=
C
e
Ω
r
+
D
r
e
. Letting Ψ
r
=
C
r
,
c
Φ
r,
(4.2) follows immediately.
To prove the second part of the
theorem, note that by assumption, for eachr
∈
{
1
,
2
}, A
^
r in (3.7) is a stability matrix. Moreover,
since A
¯
r is related to A
^
r by a similarity transformation, then A
¯
r is also a stability matrix. Now suppose Ω
r and Ψ
r satisfy (4.2) and that A
Q
r and C
Q
r satisfy (4.3). To show that the closed loop system
achieves regulation, let Φ
r
=
[
Ω
r
I
].
Using the expression for A
r
,
c in (3.4) and substituting in the expression for A
Q
r in (4.3) yields(4.6)
A
r
,
c
=
[
A
r
+
B
r
K
r
+
L
r
C
r
y
B
r
C
Q
r
+
L
r
D
r
y
+
D
r
x
B
Q
r
C
r
y
H
r
+
B
Q
r
D
r
y
]
=
[
A
r
D
r
x
0
H
r
]
+
[
B
r
0
]
[
K
r
C
Q
r
]
+
[
L
r
B
Q
r
]
[
C
r
y
D
r
y
]
. It follows that:(4.7)
A
r
,
c
Φ
r
=
[
A
r
Ω
r
+
D
r
x
H
r
]
+
[
B
r
K
r
Ω
r
+
B
r
C
Q
r
0
]
+
[
L
r
B
Q
r
]
(
C
r
y
Ω
r
+
D
r
y
)
=
[
A
r
Ω
r
+
B
r
(
K
r
Ω
r
+
C
Q
r
)
+
D
r
x
H
r
]
−
B
r
,
c
(
C
r
y
Ω
r
+
D
r
y
)
. Given that Ψ
r
=
C
Q
r
+
K
r
Ω
r from (4.3), then
based on (4.2) we have(4.8)
A
r
,
c
Φ
r
=
[
A
r
Ω
r
+
B
r
Ψ
r
+
D
r
x
H
r
]
−
B
r
,
c
(
C
r
y
Ω
r
+
D
r
y
)
=
Φ
r
H
r
−
B
r
,
c
(
C
r
y
Ω
r
+
D
r
y
)
. Therefore, (4.5) is satisfied, which implies (4.1) is satisfied and regulation can be
achieved.The
results presented above represent regulation conditions for each of the closed
loop systems in (3.7), individually. In the following sections,
regulation conditions for the system (3.7) subject to switching are discussed.


### 4.2. Equivalent Impulsive Switched Closed Loop System Model

Based on Lemma4.1, and using a
properly defined coordinate transformation, the forced switched closed loop
system (3.7) can be transformed into an unforced
impulsive switched system. To introduce the appropriate coordinate transformation,
note that the original regulation condition (4.1) can be rewritten as(4.9)
Π
^
r
H
r
=
A
^
r
Π
^
r
+
E
r
,
0
=
C
e
x
Π
^
r
+
D
r
e
, where(4.10)
Π
^
r
=
[
I
0
0
−
I
I
0
0
0
I
]
Π
r
.Consider now the following
coordinate transformation(4.11)
χ
=
χ
−
Π
^
r
w
r
. Using (4.9), the forced closed loop switched
system (3.7) can be transformed into an unforced
switched system given by(4.12)
Σ
r
cl
:
{
χ
˙
=
A
^
r
χ
,
e
=
C
e
x
χ
,
r
=
{
1
if
e
=
C
e
x
χ
≤
δ
,
2
if
e
=
C
e
x
χ
>
δ
. It should be noted from (4.11) that the coordinate transformation
varies depending on the value of r
∈
{
1
,
2
}.
Consequently, the states χ
(
t
) in the new system (4.12) undergo impulsive changes at the
switching times. Let 𝕋
s be the set of switching times τ
k, k
=
1
,
2
,
….
Therefore, for a given switching time τ
k
∈
𝕋
s,
we have that for t
=
τ
k
(4.13)
χ
(
t
+
)
=
χ
(
t
−
)
+
Δ
χ
(
t
)
,where Δ
χ
(
t
)
=
Π
^
r
(
t
−
)
w
r
(
t
−
)
(
t
)
−
Π
^
r
(
t
+
)
w
r
(
t
+
)
(
t
).
Hence, the impulsive switched closed loop system is given by(4.14)
Σ
r
cl
:
{
χ
˙
=
A
^
r
χ
,
χ
(
t
+
)
=
χ
(
t
−
)
+
Δ
χ
(
t
)
,

t
∈
T
s
,
e
=
C
e
x
χ
,
r
=
{
1
if
e
=
C
e
x
χ
≤
δ
,
2
if
e
=
C
e
x
χ
>
δ
. In the new coordinate system,
the switching surface S given in (2.2) is fixed and is expressed as follows:(4.15)
S
=
{
χ
∈
ℝ
N
:
e
=
C
e
x
χ
=
δ
}
. Using the coordinate
transformation in (4.11), the original output-feedback
regulation problem for the switched system (3.7)
is transformed into an asymptotic
stability analysis problem for the origin of the unforced impulsive switched
system (4.14).


### 4.3. Regulation Conditions for the Switched System

First, letδ
0 denote the distance between the switching
surface S and the origin. Then we have δ
0
=
|
δ
|
/
∥
C
e
x
∥.
Define(4.16)
Δ
β
m
⁡
=
max
r
∈
{
1
,
2
}
t
≥
0
‖
Π
^
r
w
r
(
t
)
‖
Let γ
=
β
+
Δ
β
m
+
ε,
where ε
≪
1,
and define the set B
γ as follows:(4.17)
B
γ
=
{
χ
∈
ℝ
N
:
‖
χ
‖
≤
γ
}
.Using the definition for χ in (4.11), and the fact that the original state
vector χ will ultimately evolve inside a bounded set B
β given by (3.11), it follows that there exists a
finite time T
γ at which χ will enter the set B
γ and continue to evolve in B
γ thereafter. A sufficient condition for
regulation is then given as follows.Theorem 4.3.
Assume that the switched closed
loop systemΣ
r
cl given in (3.7) is internally stable under arbitrary
switching, and that for each r
∈
{
1
,
2
}, Σ
r
cl given in (3.7) achieves output regulation. If(4.18)
δ
0
>
γ
, then the origin is an asymptotically
stable equilibrium point for the impulsive switched system
(4.14), implying that the switched system
(3.7) achieves regulation.Proof.
Ifδ
0
>
γ,
then the switching surface S given by (4.15) does not intersect the bounded set B
γ.
The trajectory of the state vector χ will enter the set B
γ at t
=
T
γ and continue to evolve inside B
γ thereafter. Consequently, there will be no
more switching. Since the system Σ
r
cl that is active in the halfspace containing B
γ is such that A
^
r is a stability matrix, and since 0
∈
B
γ,
the trajectory of the state vector χ asymptotically converges to the origin,
implying that regulation in the original switched closed loop system (3.7) is achieved.If condition (4.18) is not satisfied, that is, δ
0
≤
γ,
then the switching surface S must intersect with the bounded set B
γ.
To present regulation conditions in this case, define the matrices A
s
1
=
[
C
e
x
C
e
x
A
^
1
], A
s
2
=
[
C
e
x
C
e
x
A
^
2
],
and b
=
[
δ
0
].
Sufficient conditions for regulation are presented in the following theorem.Theorem 4.4.
Assume the switched closed loop
systemΣ
r
cl given in (3.7) is internally stable under arbitrary
switching and that, for each r
∈
{
1
,
2
}, Σ
r
cl given in (3.7) achieves output regulation. Moreover,
assume that
δ
0
≤
γ, rank
(
A
s
1
)
=
rank
(
A
s
2
)
=
2,
and that the direction
of vector C
e
x
→ is not parallel to that of C
e
x
A
^
1
→ and C
e
x
A
^
2
→. If the following conditions are
satisfied,(4.19)
C
e
x
(
C
e
x
A
^
r
)
T
<
0
,
b
T
(
A
s
r
A
s
r
T
)
−
1
b
>
γ
2
,
r
=
1

if
δ
>
0,
r
=
2

if
δ
<
0, or(4.20)
C
e
x
(
C
e
x
A
^
r
)
T
>
0
,
b
T
(
A
s
r
A
s
r
T
)
−
1
b
>
γ
2
,
r
=
2

if
δ
>
0,
r
=
1

if
δ
<
0, then the origin is an asymptotically
stable equilibrium point for the impulsive switched system
(4.14),
implying that the switched system (3.7) achieves regulation.Proof.
The following proof is presented for the case ofδ
>
0.
The case of δ
<
0 is treated using the same ideas. Let region 1
and region 2 denote the halfspaces where the systems Σ
1
cl and Σ
2
cl are active, respectively (see
Figures 2
and 3).
Hence, the origin is in region 1 where the system Σ
1
cl is active. Define the hypersurfaces S
v
1
=
{
χ
∈
ℝ
N
:
C
e
x
A
^
1
χ
=
0
} and S
v
2
=
{
χ
∈
ℝ
N
:
C
e
x
A
^
2
χ
=
0
}.
Then for all points in S
v
1, e
˙
=
C
e
x
A
^
1
χ
=
0 and for all points in S
v
2, e
˙
=
C
e
x
A
^
2
χ
=
0.
Since, by assumption, δ
0
≤
γ,
the switching surface S intersects with the bounded set B
γ.
Let S
γ
=
{
χ
∈
ℝ
N
:
C
e
x
χ
=
δ
and
∥
χ
∥
≤
γ
} be the intersection of the switching surface S with B
γ.
By assumption, the direction of vector C
e
x
→ is not parallel to that of C
e
x
A
^
1
→ and C
e
x
A
^
2
→.
Therefore, the switching surface S must intersect with either of the hypersurfaces S
v
1 or S
v
2.
Define the intersection of S with the hypersurfaces S
v
1 and S
v
2 as L
1
=
{
χ
∈
ℝ
N
:
C
e
x
A
^
1
χ
=
0
and
C
e
x
χ
=
δ
} and L
2
=
{
χ
∈
ℝ
N
:
C
e
x
A
^
2
χ
=
0
and
C
e
x
χ
=
δ
},
respectively. For each r
∈
{
1
,
2
},
let l
r
=
min
χ
∈
L
r
∥
χ
∥ denote the distance between the origin and the
set L
r.
Then, l
r can be obtained by solving the problem of
minimizing ∥
χ
∥ subject
to the constraint A
s
r
χ
=
b. Since, by assumption, rank
(
A
s
1
)
=
rank
(
A
s
2
)
=
2,
the solution to the
above problem is l
r
=
b
T
(
A
s
r
A
s
r
T
)
−
1
b, r
∈
{
1
,
2
}. If l
r
2
=
b
T
(
A
s
r
A
s
r
T
)
−
1
b
>
γ
2, then the sets L
r, r
∈
{
1
,
2
},
are outside the bounded set B
γ.
In this situation, the set S
γ does not intersect with the sets S
v
1 or S
v
2.
If C
e
x
(
C
e
x
A
^
1
)
T
<
0 and l
1
2
=
b
T
(
A
s
1
A
s
1
T
)
−
1
b
>
γ
2, then the set S
γ will be located
in one of the two halfspaces defined by the hypersurface S
v
1 and where
all the points satisfy e
˙
=
C
e
x
A
^
1
χ
<
0. Similarly, if C
e
x
(
C
e
x
A
^
2
)
T
>
0 and l
2
2
=
b
T
(
A
s
2
A
s
2
T
)
−
1
b
>
γ
2, then the set S
γ will be located
in one of the two halfspaces defined by the hypersurface S
v
2 and where
all the points satisfy e
˙
=
C
e
x
A
^
2
χ
>
0. By deriving conditions on e
˙,
and using the stability properties of Σ
1
cl and Σ
2
cl,
it is possible to conclude about the asymptotic stability of the origin of the
impulsive switched closed loop system (4.14).
In the following, two cases are considered in
detail.
Case 1 (C
e
x
(
C
e
x
A
^
1
)
T
<
0 and b
T
(
A
s
1
A
s
1
T
)
−
1
b
>
γ
2 (see Figure 2)).
In this case,
all the state trajectories leaving the setS
γ and entering region 1 have e
˙
<
0.
Therefore, if the state trajectory leaves the set S
γ and enters region 1, then it will not hit it
again. Two possible cases can be considered here depending on whether the state
trajectory enters the bounded set B
γ from region 1 or region 2 at time T
γ.
In this case, the state trajectory enters the set B
γ from region 1, then the state trajectory
cannot hit the switching surface S since e
˙
<
0 for all states on the switching surface within
the set B
γ.
Therefore, the state trajectory is confined to evolve in
region 1 inside the set B
γ.
Since Σ
1
cl is a stable system, the state trajectory will
converge to the origin asymptotically, which implies that regulation for the
switched system (3.7) is achieved. Consider
now the case where the state trajectory enters the set B
γ from region 2. Since the state
trajectory is confined to evolve inside the set B
γ and since the system Σ
2
cl is asymptotically stable, the state trajectory
must hit the switching surface and switching must take place. Following
switching, and based on the dynamics of the impulsive switched closed system as
presented in (4.14), the state vector immediately
following switching will be in region 1. Since e
˙
<
0 for all states on the switching surface within
the set B
γ for the system Σ
1
cl,
the state trajectory cannot cross the switching surface again and will continue
to evolve in region 1 thereafter. The state trajectory will approach the origin
asymptotically, which implies that
regulation will be achieved.
Case 2 (C
e
x
(
C
e
x
A
^
2
)
T
>
0 and b
T
(
A
s
2
A
s
2
T
)
−
1
b
>
γ
2 (see Figure 3)).
In this case,
all the state trajectories leaving the setS
γ and entering region 2 have e
˙
>
0.
Assume the state trajectory enters the set B
γ from region 2 at the time T
γ.
Given that the system Σ
2
cl is asymptotically stable, and that the
trajectories are confined to evolve inside B
γ,
the state trajectory must cross the switching surface to approach the origin.
However, this is not possible since, with respect to the system Σ
2
cl, e
˙
>
0 for all χ
∈
S
γ.
Therefore, it is not possible to have the state trajectory enter the set B
γ from region 2 at time T
γ.
Consider now the case, where the state trajectory enters the set B
γ from region 1. If the state trajectory crosses
the switching surface to enter into region 2, then that will result in a
contradiction similar to that discussed for the case, where the state
trajectory enters the set B
γ from region 2. Therefore, once the state
trajectory enters the set B
γ from region 1, it will continue to evolve in
region 1 forever and will never cross the switching surface to enter into region
2 for t
>
T
γ.
Since Σ
1
cl is a stable system, then the state trajectory
will converge to the origin asymptotically, which implies that regulation for
the switched system (3.7) is achieved.
Therefore, based on the above analysis, if the conditions for
Case1 given by (4.19) or the
conditions for Case 2 given by (4.20) are satisfied, then regulation in the
switched system (3.7) can be achieved.Figure 2
State
trajectories ofΣ
1
cl with initial conditions on the switching
surface inside the set B
γ for Case 1.Figure 3
State
trajectories ofΣ
2
cl with initial conditions on the switching
surface inside the set B
γ for Case 2.


## 4.1. Regulation Conditions forΣ
1
cland Σ
2
cl

LetA
¯
r
=
[
A
r
B
r
C
r
,
c
B
r
,
c
C
r
y
A
r
,
c
] and E
¯
r
=
[
D
r
x
B
r
,
c
D
r
y
] in (2.4). For each of the systems Σ
r, r
∈
{
1
,
2
},
and using a controller of the form (2.3), regulation conditions are given by
the following lemma.Lemma 4.1 (see [15]).
For eachr
∈
{
1
,
2
},
consider the system Σ
r in (2.1) and a controller ℂ
r in (2.3), and assume that A
¯
r is a stability matrix. Then, for each r
∈
{
1
,
2
}, Σ
r
cl in (2.4) achieves regulation if and only if
there exists a matrix Π
r that solves the linear matrix equations:(4.1)
Π
r
H
r
=
A
¯
r
Π
r
+
E
¯
r
,
0
=
C
e
x
Π
r
+
D
r
e
.
Similarly,
using the parameterized feedback regulator ℂ
r given in (2.3) and (3.4), regulation conditions for the
switched closed loop system (3.7)
can be derived and are presented in the following theorem.Theorem 4.2.
Assume
that for eachr
∈
{
1
,
2
},
the closed loop system Σ
r
cl given in (3.7) is exponentially stable. Then each of
the systems Σ
1
cl and Σ
2
cl achieves regulation only if,
for each r
∈
{
1
,
2
},
there exists a pair of matrices (
Ω
r
,
Ψ
r
) which satisfy the following equations:(4.2)
Ω
r
H
r
=
A
r
Ω
r
+
B
r
Ψ
r
+
D
r
x
,
0
=
C
e
Ω
r
+
D
r
e
. Furthermore,
if the linear matrix equations in (4.2) admit a solution (
Ω
r
,
Ψ
r
),
and the matrices A
Q
r and C
Q
r in (3.3) are taken to be of the form(4.3)
A
Q
r
=
H
r
+
B
Q
r
D
r
y
,
C
Q
r
=
Ψ
r
−
K
r
Ω
r
, then,
for each r
∈
{
1
,
2
}, Σ
r
cl in (3.7) achieves regulation.Proof.
Equation
(4.1) can be rewritten as(4.4)
Π
r
H
r
=
[
A
r
B
r
C
r
,
c
B
r
,
c
C
r
y
A
r
,
c
]
Π
r
+
[
D
r
x
B
r
,
c
D
r
y
]
,
0
=
[
C
e
0
]
Π
r
+
D
r
e
. Partition Π
r as Π
r
=
[
Ω
r
Φ
r
].
Then (4.1) is the same as(4.5)
Ω
r
H
r
=
A
r
Ω
r
+
B
r
C
r
,
c
Φ
r
+
D
r
x
,
Φ
r
H
r
=
A
r
,
c
Φ
r
+
B
r
,
c
(
C
r
y
Ω
r
+
D
r
y
)
,
0
=
C
e
Ω
r
+
D
r
e
. Letting Ψ
r
=
C
r
,
c
Φ
r,
(4.2) follows immediately.
To prove the second part of the
theorem, note that by assumption, for eachr
∈
{
1
,
2
}, A
^
r in (3.7) is a stability matrix. Moreover,
since A
¯
r is related to A
^
r by a similarity transformation, then A
¯
r is also a stability matrix. Now suppose Ω
r and Ψ
r satisfy (4.2) and that A
Q
r and C
Q
r satisfy (4.3). To show that the closed loop system
achieves regulation, let Φ
r
=
[
Ω
r
I
].
Using the expression for A
r
,
c in (3.4) and substituting in the expression for A
Q
r in (4.3) yields(4.6)
A
r
,
c
=
[
A
r
+
B
r
K
r
+
L
r
C
r
y
B
r
C
Q
r
+
L
r
D
r
y
+
D
r
x
B
Q
r
C
r
y
H
r
+
B
Q
r
D
r
y
]
=
[
A
r
D
r
x
0
H
r
]
+
[
B
r
0
]
[
K
r
C
Q
r
]
+
[
L
r
B
Q
r
]
[
C
r
y
D
r
y
]
. It follows that:(4.7)
A
r
,
c
Φ
r
=
[
A
r
Ω
r
+
D
r
x
H
r
]
+
[
B
r
K
r
Ω
r
+
B
r
C
Q
r
0
]
+
[
L
r
B
Q
r
]
(
C
r
y
Ω
r
+
D
r
y
)
=
[
A
r
Ω
r
+
B
r
(
K
r
Ω
r
+
C
Q
r
)
+
D
r
x
H
r
]
−
B
r
,
c
(
C
r
y
Ω
r
+
D
r
y
)
. Given that Ψ
r
=
C
Q
r
+
K
r
Ω
r from (4.3), then
based on (4.2) we have(4.8)
A
r
,
c
Φ
r
=
[
A
r
Ω
r
+
B
r
Ψ
r
+
D
r
x
H
r
]
−
B
r
,
c
(
C
r
y
Ω
r
+
D
r
y
)
=
Φ
r
H
r
−
B
r
,
c
(
C
r
y
Ω
r
+
D
r
y
)
. Therefore, (4.5) is satisfied, which implies (4.1) is satisfied and regulation can be
achieved.The
results presented above represent regulation conditions for each of the closed
loop systems in (3.7), individually. In the following sections,
regulation conditions for the system (3.7) subject to switching are discussed.


## 4.2. Equivalent Impulsive Switched Closed Loop System Model

Based on Lemma4.1, and using a
properly defined coordinate transformation, the forced switched closed loop
system (3.7) can be transformed into an unforced
impulsive switched system. To introduce the appropriate coordinate transformation,
note that the original regulation condition (4.1) can be rewritten as(4.9)
Π
^
r
H
r
=
A
^
r
Π
^
r
+
E
r
,
0
=
C
e
x
Π
^
r
+
D
r
e
, where(4.10)
Π
^
r
=
[
I
0
0
−
I
I
0
0
0
I
]
Π
r
.Consider now the following
coordinate transformation(4.11)
χ
=
χ
−
Π
^
r
w
r
. Using (4.9), the forced closed loop switched
system (3.7) can be transformed into an unforced
switched system given by(4.12)
Σ
r
cl
:
{
χ
˙
=
A
^
r
χ
,
e
=
C
e
x
χ
,
r
=
{
1
if
e
=
C
e
x
χ
≤
δ
,
2
if
e
=
C
e
x
χ
>
δ
. It should be noted from (4.11) that the coordinate transformation
varies depending on the value of r
∈
{
1
,
2
}.
Consequently, the states χ
(
t
) in the new system (4.12) undergo impulsive changes at the
switching times. Let 𝕋
s be the set of switching times τ
k, k
=
1
,
2
,
….
Therefore, for a given switching time τ
k
∈
𝕋
s,
we have that for t
=
τ
k
(4.13)
χ
(
t
+
)
=
χ
(
t
−
)
+
Δ
χ
(
t
)
,where Δ
χ
(
t
)
=
Π
^
r
(
t
−
)
w
r
(
t
−
)
(
t
)
−
Π
^
r
(
t
+
)
w
r
(
t
+
)
(
t
).
Hence, the impulsive switched closed loop system is given by(4.14)
Σ
r
cl
:
{
χ
˙
=
A
^
r
χ
,
χ
(
t
+
)
=
χ
(
t
−
)
+
Δ
χ
(
t
)
,

t
∈
T
s
,
e
=
C
e
x
χ
,
r
=
{
1
if
e
=
C
e
x
χ
≤
δ
,
2
if
e
=
C
e
x
χ
>
δ
. In the new coordinate system,
the switching surface S given in (2.2) is fixed and is expressed as follows:(4.15)
S
=
{
χ
∈
ℝ
N
:
e
=
C
e
x
χ
=
δ
}
. Using the coordinate
transformation in (4.11), the original output-feedback
regulation problem for the switched system (3.7)
is transformed into an asymptotic
stability analysis problem for the origin of the unforced impulsive switched
system (4.14).


## 4.3. Regulation Conditions for the Switched System

First, letδ
0 denote the distance between the switching
surface S and the origin. Then we have δ
0
=
|
δ
|
/
∥
C
e
x
∥.
Define(4.16)
Δ
β
m
⁡
=
max
r
∈
{
1
,
2
}
t
≥
0
‖
Π
^
r
w
r
(
t
)
‖
Let γ
=
β
+
Δ
β
m
+
ε,
where ε
≪
1,
and define the set B
γ as follows:(4.17)
B
γ
=
{
χ
∈
ℝ
N
:
‖
χ
‖
≤
γ
}
.Using the definition for χ in (4.11), and the fact that the original state
vector χ will ultimately evolve inside a bounded set B
β given by (3.11), it follows that there exists a
finite time T
γ at which χ will enter the set B
γ and continue to evolve in B
γ thereafter. A sufficient condition for
regulation is then given as follows.Theorem 4.3.
Assume that the switched closed
loop systemΣ
r
cl given in (3.7) is internally stable under arbitrary
switching, and that for each r
∈
{
1
,
2
}, Σ
r
cl given in (3.7) achieves output regulation. If(4.18)
δ
0
>
γ
, then the origin is an asymptotically
stable equilibrium point for the impulsive switched system
(4.14), implying that the switched system
(3.7) achieves regulation.Proof.
Ifδ
0
>
γ,
then the switching surface S given by (4.15) does not intersect the bounded set B
γ.
The trajectory of the state vector χ will enter the set B
γ at t
=
T
γ and continue to evolve inside B
γ thereafter. Consequently, there will be no
more switching. Since the system Σ
r
cl that is active in the halfspace containing B
γ is such that A
^
r is a stability matrix, and since 0
∈
B
γ,
the trajectory of the state vector χ asymptotically converges to the origin,
implying that regulation in the original switched closed loop system (3.7) is achieved.If condition (4.18) is not satisfied, that is, δ
0
≤
γ,
then the switching surface S must intersect with the bounded set B
γ.
To present regulation conditions in this case, define the matrices A
s
1
=
[
C
e
x
C
e
x
A
^
1
], A
s
2
=
[
C
e
x
C
e
x
A
^
2
],
and b
=
[
δ
0
].
Sufficient conditions for regulation are presented in the following theorem.Theorem 4.4.
Assume the switched closed loop
systemΣ
r
cl given in (3.7) is internally stable under arbitrary
switching and that, for each r
∈
{
1
,
2
}, Σ
r
cl given in (3.7) achieves output regulation. Moreover,
assume that
δ
0
≤
γ, rank
(
A
s
1
)
=
rank
(
A
s
2
)
=
2,
and that the direction
of vector C
e
x
→ is not parallel to that of C
e
x
A
^
1
→ and C
e
x
A
^
2
→. If the following conditions are
satisfied,(4.19)
C
e
x
(
C
e
x
A
^
r
)
T
<
0
,
b
T
(
A
s
r
A
s
r
T
)
−
1
b
>
γ
2
,
r
=
1

if
δ
>
0,
r
=
2

if
δ
<
0, or(4.20)
C
e
x
(
C
e
x
A
^
r
)
T
>
0
,
b
T
(
A
s
r
A
s
r
T
)
−
1
b
>
γ
2
,
r
=
2

if
δ
>
0,
r
=
1

if
δ
<
0, then the origin is an asymptotically
stable equilibrium point for the impulsive switched system
(4.14),
implying that the switched system (3.7) achieves regulation.Proof.
The following proof is presented for the case ofδ
>
0.
The case of δ
<
0 is treated using the same ideas. Let region 1
and region 2 denote the halfspaces where the systems Σ
1
cl and Σ
2
cl are active, respectively (see
Figures 2
and 3).
Hence, the origin is in region 1 where the system Σ
1
cl is active. Define the hypersurfaces S
v
1
=
{
χ
∈
ℝ
N
:
C
e
x
A
^
1
χ
=
0
} and S
v
2
=
{
χ
∈
ℝ
N
:
C
e
x
A
^
2
χ
=
0
}.
Then for all points in S
v
1, e
˙
=
C
e
x
A
^
1
χ
=
0 and for all points in S
v
2, e
˙
=
C
e
x
A
^
2
χ
=
0.
Since, by assumption, δ
0
≤
γ,
the switching surface S intersects with the bounded set B
γ.
Let S
γ
=
{
χ
∈
ℝ
N
:
C
e
x
χ
=
δ
and
∥
χ
∥
≤
γ
} be the intersection of the switching surface S with B
γ.
By assumption, the direction of vector C
e
x
→ is not parallel to that of C
e
x
A
^
1
→ and C
e
x
A
^
2
→.
Therefore, the switching surface S must intersect with either of the hypersurfaces S
v
1 or S
v
2.
Define the intersection of S with the hypersurfaces S
v
1 and S
v
2 as L
1
=
{
χ
∈
ℝ
N
:
C
e
x
A
^
1
χ
=
0
and
C
e
x
χ
=
δ
} and L
2
=
{
χ
∈
ℝ
N
:
C
e
x
A
^
2
χ
=
0
and
C
e
x
χ
=
δ
},
respectively. For each r
∈
{
1
,
2
},
let l
r
=
min
χ
∈
L
r
∥
χ
∥ denote the distance between the origin and the
set L
r.
Then, l
r can be obtained by solving the problem of
minimizing ∥
χ
∥ subject
to the constraint A
s
r
χ
=
b. Since, by assumption, rank
(
A
s
1
)
=
rank
(
A
s
2
)
=
2,
the solution to the
above problem is l
r
=
b
T
(
A
s
r
A
s
r
T
)
−
1
b, r
∈
{
1
,
2
}. If l
r
2
=
b
T
(
A
s
r
A
s
r
T
)
−
1
b
>
γ
2, then the sets L
r, r
∈
{
1
,
2
},
are outside the bounded set B
γ.
In this situation, the set S
γ does not intersect with the sets S
v
1 or S
v
2.
If C
e
x
(
C
e
x
A
^
1
)
T
<
0 and l
1
2
=
b
T
(
A
s
1
A
s
1
T
)
−
1
b
>
γ
2, then the set S
γ will be located
in one of the two halfspaces defined by the hypersurface S
v
1 and where
all the points satisfy e
˙
=
C
e
x
A
^
1
χ
<
0. Similarly, if C
e
x
(
C
e
x
A
^
2
)
T
>
0 and l
2
2
=
b
T
(
A
s
2
A
s
2
T
)
−
1
b
>
γ
2, then the set S
γ will be located
in one of the two halfspaces defined by the hypersurface S
v
2 and where
all the points satisfy e
˙
=
C
e
x
A
^
2
χ
>
0. By deriving conditions on e
˙,
and using the stability properties of Σ
1
cl and Σ
2
cl,
it is possible to conclude about the asymptotic stability of the origin of the
impulsive switched closed loop system (4.14).
In the following, two cases are considered in
detail.
Case 1 (C
e
x
(
C
e
x
A
^
1
)
T
<
0 and b
T
(
A
s
1
A
s
1
T
)
−
1
b
>
γ
2 (see Figure 2)).
In this case,
all the state trajectories leaving the setS
γ and entering region 1 have e
˙
<
0.
Therefore, if the state trajectory leaves the set S
γ and enters region 1, then it will not hit it
again. Two possible cases can be considered here depending on whether the state
trajectory enters the bounded set B
γ from region 1 or region 2 at time T
γ.
In this case, the state trajectory enters the set B
γ from region 1, then the state trajectory
cannot hit the switching surface S since e
˙
<
0 for all states on the switching surface within
the set B
γ.
Therefore, the state trajectory is confined to evolve in
region 1 inside the set B
γ.
Since Σ
1
cl is a stable system, the state trajectory will
converge to the origin asymptotically, which implies that regulation for the
switched system (3.7) is achieved. Consider
now the case where the state trajectory enters the set B
γ from region 2. Since the state
trajectory is confined to evolve inside the set B
γ and since the system Σ
2
cl is asymptotically stable, the state trajectory
must hit the switching surface and switching must take place. Following
switching, and based on the dynamics of the impulsive switched closed system as
presented in (4.14), the state vector immediately
following switching will be in region 1. Since e
˙
<
0 for all states on the switching surface within
the set B
γ for the system Σ
1
cl,
the state trajectory cannot cross the switching surface again and will continue
to evolve in region 1 thereafter. The state trajectory will approach the origin
asymptotically, which implies that
regulation will be achieved.
Case 2 (C
e
x
(
C
e
x
A
^
2
)
T
>
0 and b
T
(
A
s
2
A
s
2
T
)
−
1
b
>
γ
2 (see Figure 3)).
In this case,
all the state trajectories leaving the setS
γ and entering region 2 have e
˙
>
0.
Assume the state trajectory enters the set B
γ from region 2 at the time T
γ.
Given that the system Σ
2
cl is asymptotically stable, and that the
trajectories are confined to evolve inside B
γ,
the state trajectory must cross the switching surface to approach the origin.
However, this is not possible since, with respect to the system Σ
2
cl, e
˙
>
0 for all χ
∈
S
γ.
Therefore, it is not possible to have the state trajectory enter the set B
γ from region 2 at time T
γ.
Consider now the case, where the state trajectory enters the set B
γ from region 1. If the state trajectory crosses
the switching surface to enter into region 2, then that will result in a
contradiction similar to that discussed for the case, where the state
trajectory enters the set B
γ from region 2. Therefore, once the state
trajectory enters the set B
γ from region 1, it will continue to evolve in
region 1 forever and will never cross the switching surface to enter into region
2 for t
>
T
γ.
Since Σ
1
cl is a stable system, then the state trajectory
will converge to the origin asymptotically, which implies that regulation for
the switched system (3.7) is achieved.
Therefore, based on the above analysis, if the conditions for
Case1 given by (4.19) or the
conditions for Case 2 given by (4.20) are satisfied, then regulation in the
switched system (3.7) can be achieved.Figure 2
State
trajectories ofΣ
1
cl with initial conditions on the switching
surface inside the set B
γ for Case 1.Figure 3
State
trajectories ofΣ
2
cl with initial conditions on the switching
surface inside the set B
γ for Case 2.


## 5. Regulator Synthesis for the Switched System

Based on the regulation
conditions for the switched system proposed in the previous section, a regulator
synthesis approach is presented in this section. The proposed synthesis approach
is based on solving a set of properly formulated bilinear matrix inequalities
(BMIs). The main idea behind the regulator synthesis approach is as follows.
Consider an output-feedback controllerℂ
r as given in (2.3) and (3.4), where A
Q
r
=
H
r
+
B
Q
r
D
r
y and C
Q
r
=
Ψ
r
−
K
r
Ω
r.
Since (
Ω
r
,
Ψ
r
) is the solution to the Sylvester equation (4.2), ℂ
r is only parameterized in the unknown matrices K
r, L
r,
and B
Q
r.
Assume ℂ
r is such that the resulting closed system
satisfies (3.9). Then, based on Lemma 3.2 and Theorem 4.2,
the switched closed loop system Σ
r
cl given in (3.7) is internally stable under arbitrary
switching and, for each r
∈
{
1
,
2
}, Σ
r
cl given in (3.7) also achieves output regulation. In
this case, if any of (4.18), (4.19), and (4.20) is satisfied, then
based on
Theorems
4.3 and 4.4, regulation in the switched system (3.7) is achieved. The set of conditions in
Lemma 3.2 and Theorems 4.3
and 4.4 yield bilinear matrix inequalities in the unknown
parameters K
r, L
r,
and B
Q
r.
In the following, a solution procedure for the formulated BMIs is proposed to
determine the unknown parameters K
r, L
r,
and B
Q
r in ℂ
r.The
sufficient regulation condition (4.18) given in Theorem 4.3 is equivalent to(5.1)
β
<
δ
0
−
Δ
β
m
−
ε
.Using β
=
c
w
m
/
α and letting κ
=
(
δ
0
−
Δ
β
m
−
ε
)
/
w
m,
condition (4.18) is equivalent to(5.2)
c
α
<
κ
. If (5.2) cannot be satisfied, the switching
surface S intersects the set B
γ and conditions (4.19) and (4.20) in
Theorem 4.4 need to be verified. For
the switched closed loop system (3.7), the parameters c and α can be estimated using the following matrix
inequalities [22]:(5.3)
A
^
r
T
P
+
P
A
^
r
+
2
α
P
<
0
,

α
>
0
,
r
∈
{
1
,
2
}
,
I
<
P
<
c
2
I
,

c
>
0. If the matrices K
r, L
r, B
Q
r,
and C
Q
r in A
^
r are unknown, then the above equations define a
bilinear matrix inequality. Combining (5.2) and (5.3), the synthesis procedure for the
controller ℂ
r can be realized by solving the following BMIs:(5.4)
[
A
r
+
B
r
K
r
B
r
K
r
B
r
Ψ
r
−
B
r
K
r
Ω
r
0
A
r
+
L
r
C
r
y
D
r
x
+
L
r
D
r
y
0
B
Q
r
C
r
y
H
r
+
B
Q
r
D
r
y
]
T
P
+
P
[
A
r
+
B
r
K
r
B
r
K
r
B
r
Ψ
r
−
B
r
K
r
Ω
r
0
A
r
+
L
r
C
r
y
D
r
x
+
L
r
D
r
y
0
B
Q
r
C
r
y
H
r
+
B
Q
r
D
r
y
]


+
2
α
P
<
0
,

α
>
0
,
r
∈
{
1
,
2
}
,
(5.5)
I
<
P
<
c
2
I
,

c
>
0
,
(5.6)
c
α
<
κ
, where the unknown parameters are K
r, L
r, B
Q
r, P, α,
and c.
Once K
r, L
r, B
Q
r are determined based on the above BMIs, the
matrices A
Q
r and C
Q
r can be calculated using (4.3) as given in
Theorem 4.2.In this paper, a regulator synthesis algorithm, referred to
as theP-ℂ  iteration
algorithm, will be used to find a
parameterized regulator that satisfies either one of the regulation conditions
(4.18), (4.19), or (4.20). The P-ℂ
algorithm iteratively solves for the Lyapunov
matrix P and the parameters in the controller ℂ
r given in (2.3). The basic idea of the P-ℂ iteration is that a BMI can be converted into
an LMI when some of the parameters in the BMI are fixed. The approach for
solving BMI problems is to alternate between two optimization problems subject
to LMIs, which are related to the matrix P in (5.4) and (5.5) and the parameters in the controller ℂ
r in (2.3), respectively. In the
algorithm, the
input data is represented by A
r, B
r, C
r
y, C
e, D
r
x, D
r
y, D
r
e, H
r, δ,
and w
m,
whereas the unknown variables to be determined are K
r, L
r, B
Q
r, A
Q
r, C
Q
r,
P, α,
and c.
The algorithm is summarized below, where K
r
i, L
r
i, B
Q
r
i, P
i, α
i,
and c
i denote the solutions K
r, L
r, B
Q
r, P, α,
and c obtained at the  ith iteration of the algorithm.(1)
Calculating
κ: Determine (
Ω
r
,
Ψ
r
) by solving the Sylvester equation (4.2) and let Π
r
=
[
Ω
r
Φ
r
] with Φ
r
=
[
Ω
r
I
].
Let Π
^
r be as in (4.10). Determine Δ
β
m using (4.16),
then calculate
κ
=
(
δ
0
−
Δ
β
m
−
ε
)
/
w
m.
(2)
Initializing
ℂ
r: Initialize the
controller parameters K
r
0, L
r
0,
and B
Q
r
0 to make the switched closed loop system (3.7) internally stable, which can be
realized by solving the following two LMIs separately for the unknown matrices K
r, L
r, B
Q
r, P
K,
and P
L with preset constants α
K
≥
0 and α
L
≥
0, (5.7)
[
A
r
+
B
r
K
r
]
T
P
K
+
P
K
[
A
r
+
B
r
K
r
]
+
α
K
P
K
<
0
,

r
∈
{
1
,
2
}
,
[
A
r
+
L
r
C
r
y
D
r
x
+
L
r
D
r
y
B
Q
r
C
r
y
H
r
+
B
Q
r
D
r
y
]
T
P
L
+
P
L
[
A
r
+
L
r
C
r
y
D
r
x
+
L
r
D
r
y
B
Q
r
C
r
y
H
r
+
B
Q
r
D
r
y
]
+
α
L
P
L
<
0
,

r
∈
{
1
,
2
}
.
(3)
Initializing
α
0:
Based on the initial controller
parameters K
r
0, L
r
0,
and B
Q
r
0 from step 2, initialize the maximum decay rate α
0 by solving the following optimization problem:(5.8)
max
⁡
α
subject
to
(
5.4
)
.
(4)
P-Step: At the ith iteration, i
=
1
,
2
,
…,
given the matrices K
r
i
−
1, L
r
i
−
1,
and B
Q
r
i
−
1 and the scalar α
i
−
1,
solve the following optimization problem for P
i and c
i:(5.9)
min
⁡
c
subject
to
(
5.4
)
and
⁢
⁢
(
5.5
)
.
(5)
ℂ-Step: At
the ith iteration, i
=
1
,
2
,
…,
given the matrix P
i and c
i from step 4, solve the following optimization problem
for K
r
i, L
r
i, B
Q
r
i,
and α
i:(5.10)
max
⁡
α
subject
to
(
5.4
)
and
⁢
⁢
(
5.5
)
.
(6)
Verification: Verify the
constraints (5.6), (4.19), and (4.20), and if any of the three conditions is
satisfied, compute A
Q
r and C
Q
r using (4.3), then stop the
algorithm. If none of (5.6),
(4.19), and (4.20) is satisfied, then go to step 4.The iterative loop is
repeatedly executed until a solution is found, or there is no major reduction
inc
/
α relative to the previous iteration. The algorithm converges
to a local solution of c
/
α,
since at each ith iteration of the algorithm, we have c
i
≤
c
i
−
1 and α
i
≥
α
i
−
1.
Therefore, if
no solution can satisfy the regulation conditions (5.6), (4.19), or (4.20) by the iterative procedure described above, then the
initial parameters K
r
0, L
r
0,
and B
Q
r
0 can be adjusted by changing the decay rates α
K and α
L in (5.7), and restarting the iterative
procedure again. If the above algorithm  yields a solution, then, according to
Theorems 4.3 or 4.4, the switched closed loop system will achieve regulation.


## 6. Numerical Example

In this section, the regulator
synthesis method proposed in this paper will be used to design a controller
that cancels the contact vibrations in a mechanical system.
Figure4 shows the
diagram of such a system consisting of a mass m,
a contact surface S
c,
and their respective coordinates. The mass m is attached to a spring with stiffness k and a damper with damping coefficient c.
The mass m moves
only in the vertical direction, whereas the contact surface underneath it moves
to the left. The mass m may enter into intermittent contact with the
surface S
c,
resulting in contact vibrations. When the mass m enters into contact with the surface S
c,
the contact characteristics are represented by a spring with stiffness k
c and a damper with damping coefficient c
c.
The force F represents the external force used to control the mass m,
whereas F
a represents a disturbance force. This model can
be found in many applications, such as the interface between the
read/write head and the disk surface in hard disk drive systems. In the
following, and with respect to the system shown in Figure 4, the
control objective is for the mass m to follow the displacement of the contact
surface S
c,
while maintaining a desired constant separation in the vertical direction. Let ν be the deviation of the mass from its
equilibrium position. Then the equations of motion of the mass can be written
as(6.1)
m
ν
¨
+
c
ν
˙
+
k
ν
=
−
F
+
F
a
,

when
ν
>
ν
s
⁢
(
noncontact
mode
)
,
m
ν
¨
+
c
ν
˙
+
k
ν
=
−
F
+
F
a
+
F
c
,

when
ν
≤
ν
s
(
contact
mode
)
, where F
c is the contact force expressed as(6.2)
F
c
=
c
c
(
ν
˙
s
−
ν
˙
)
+
k
c
(
ν
s
−
ν
)
,and where ν
s is the displacement of the contact surface.
The wavy contact surface profile is expressed as a linear combination of
sinusoidal functions:(6.3)
ν
s
=
∑
k
c
k
cos
⁡
(
ω
k
t
+
ϕ
k
)
+
ν
e
, with amplitudes c
k,
frequencies ω
k,
phases ϕ
k, k
=
1
,
2
,
…,
and constant offset ν
e.
Therefore, the distance between the mass m and contact surface S
c is ν
−
ν
s.
The control signal is defined as u
=
−
F
/
m for the noncontact situation and u
=
−
(
F
−
k
c
ν
e
)
/
m,
when contact takes place. Let −
ν
e denote the desired distance between the mass m and contact surface S
c.
The output y,
to be fed to the controller, is defined as(6.4)
y
=
ν
.The performance variable e is defined to be the difference between the
actual distance ν
−
ν
s and the desired distance −
ν
e,(6.5)
e
=
ν
−
ν
s
+
ν
e
.Therefore, the system will
switch between the contact and noncontact modes according to the value of the
performance variable e.
If e
>
δ
=
ν
e,
the system will operate in the noncontact mode, and if e
≤
δ
=
ν
e,
the system will operate in the contact mode. Let x
1
=
ν and x
2
=
ν
˙,
then the switched system model is given by(6.6)
Σ
r
:
{
[
x
˙
1
x
˙
2
]
=
A
r
[
x
1
x
2
]
+
[
0
1
]
u
+
[
0
0
1
0
]
d
r
y
=
[
1
0
]
[
x
1
x
2
]
e
=
[
1
0
]
[
x
1
x
2
]
+
[
0
−
1
]
d
r
r
=
{
1
if
e
≤
ν
e
,
2
if
e
>
ν
e
, where(6.7)
A
1
=
[
0
1
−
(
k
+
k
c
)
m
−
(
c
+
c
c
)
m
]
,


d
1
=
[
1
m
(
c
c
ν
˙
s
+
k
c
ν
s
−
k
c
ν
e
+
F
a
)
ν
s
−
ν
e
]
,
A
2
=
[
0
1
−
k
m
−
c
m
]
,


d
2
=
[
F
a
/
m
ν
s
−
ν
e
]
.
Consider
a model for the read/write head
and disk
surface interface in hard disk drive systems similar to that in [11, 12]. The surface
profile is given by v
s
=
(
sin
(
120
π
t
)
−
30
)
×
10
−
6
m,
and the force F
a
=
(
1.5
sin
(
120
π
t
+
π
)
+
0.1
cos
(
120
π
t
+
π
)
)
×
10
−
3
N.
Therefore ， we have δ
=
v
e
=
−
30
×
10
−
6
m. Let m, k, c, k
c,
and c
c be 200 mg, 1.5
×
10
3 N/m, 0.05 N/m/Sec, 1.5
×
10
4 N/m, and 0.3 N/m/Sec,
respectively. In the following, a regulator for the switched system is designed
using the synthesis procedure described in the previous section. Simulation
results will illustrate the performance of the proposed regulator in
maintaining the desired system output despite the presence of switching.Figure 4
Diagram of a
mechanical system with switched dynamics.In
order to find a feasible solution for the BMIs, we introduce a coordinate
transformation as[
x
1
′
x
2
′
]
=
T
[
x
1
x
2
],
where T
=
[
0.25
0.25
×
10
−
8
0.50
−
0.50
×
10
−
8
].
Rewriting (6.6) in the state space form (2.1) results in(6.8)
A
1
=
[
49999991
−
24999995
100000017
−
50000008
]
,
A
2
=
[
49999998
−
24999999
100000002
−
50000001
]
,
H
1
=
H
2
=
[
0
120
π
−
120
π
0
]
,

w
10
=
w
20
=
[
0
1
×
10
−
6
]
,

B
1
=
B
2
=
[
0.25
×
10
−
8
−
0.50
×
10
−
8
]
,
C
1
y
=
C
2
y
=
C
e
=
[
2
1
]
,
D
1
x
=
[
0.21
1.41
×
10
−
3
−
0.41
−
2.82
×
10
−
3
]
,

D
2
x
=
[
1
.87
×
10
−
2
1.25
×
10
−
3
−
3.75
×
10
−
2
−
2.50
×
10
−
3
]
,

D
1
y
=
D
2
y
=
0
,
D
1
e
=
D
2
e
=
[
−
1
0
]
.      It is desired to design a
regulator that can reject the disturbance in the switched system. Based on
Theorem 4.2, solving the Sylvester equation (4.2) yields(6.9)
Ψ
1
=
[
−
14.21
9.42
]
×
10
4
,


Ψ
2
=
[
−
14.21
−
40.58
]
×
10
4
,
Ω
1
=
Ω
2
=
[
0.25
9.42
×
10
−
7
0.50
−
1.88
×
10
−
6
]
,


Π
1
=
Π
2
=
[
Ω
1
Ω
1
I
2
×
2
]
. Based on (4.16), we obtain Δ
β
m
=
1.76
×
10
−
6.
Since δ
=
−
30
×
10
−
6m and w
m
=
0.62
×
10
−
6,
and with ε
=
0.01,
we have κ
=
18.81 in (5.2). Using the P-ℂ   algorithm proposed above, we obtain(6.10)
K
1
=
[
−
4.20
1.88
]
×
10
15
,


K
2
=
[
−
8.98
4.02
]
×
10
15
,
L
1
=
[
−
1.51
−
3.02
]
×
10
7
,


L
2
=
[
−
0.52
−
1.04
]
×
10
7
,
B
Q
1
=
[
−
29.69
4.34
]
×
10
7
,


B
Q
2
=
[
−
5.21
3.91
]
×
10
7
,


c
α
=
23.61.     Then, based on (4.3), we have(6.11)
C
Q
1
=
[
11006.43
7.51
]
×
10
4
,


C
Q
2
=
[
23532.54
1.63
]
×
10
4
. Since c
/
α
=
23.61
>
κ
=
18.81,
condition (5.6) is not satisfied. Then conditions (4.19) and (4.20) will be verified. First, we obtain γ
=
c
w
m
/
α
+
Δ
β
m
+
ε
=
16.41
×
10
−
6.
Hence, after a long enough time, the trajectory will enter into the bounded set B
γ with γ
=
16.41
×
10
−
6 and continue to evolve thereafter. Given that δ
<
0 and that(6.12)
C
e
x
(
C
e
x
A
^
1
)
T
=
3
×
10
8
>
0
,
b
T
(
A
s
1
A
s
1
T
)
−
1
b
=
281.25
×
10
−
12
>
γ
2
=
268.91
×
10
−
12
, it follows that regulation
condition (4.20) in Theorem 4.4 is satisfied. Therefore,
regulation can be achieved in the switched closed loop system using the
designed controller.The
simulation results of the response of the closed loop system under switching
are illustrated in Figures (5-6). It
can be seen that if the performance variable is smaller than −
30 micrometers, the
mass enters into contact with the contact surface S
c and the model of the system switches. It can
also be seen that the disturbance d
r changes at the switching times. But even in
the presence of switching, the switched system performance variable e still converges to zero, which means the mass m asymptotically follows the contact surface S
c at the desired separation 30 micrometers.Simulation results for the case ofν
e
=
−
30 micrometers showing the performance variable e and the switching component d
r
(
1
,
1
) in the disturbance d
r.
(a)
(b)Figure 6
Simulation results for the case ofν
e
=
−
30 micrometers showing the regulated mass height v and the contact surface displacement ν
s.


## 7. Conclusion

The problem of regulation in
switched bimodal systems against known disturbance or reference signals is
discussed. A regulator design approach based on the parameterization of a set
of controllers that can achieve regulation for the switched system is
presented. The forced switched closed loop system involving the parameterized
controller is transformed into an unforced impulsive switched system.
Consequently, the original regulation problem is transformed into a stability
analysis problem for the origin of the impulsive switched system. Sufficient
conditions for regulation and a regulator synthesis method based on solving a
set of bilinear matrix inequalities are presented. A simulation example is used
to illustrate the effectiveness of the proposed regulator.



---

*Source: 341720-2008-04-17.xml*