International Journal of Automation and Computing
14(2), April 2017, 213-220
DOI: 10.1007/s11633-016-1003-5
Regional Stability of Positive Switched Linear Systems
with Multi-equilibrium Points
Zhi Liu
Yu-Zhen Wang
School of Control Science and Engineering, Shandong University, Ji nan 250061, China
Abstract: This paper studies the regional stability for positive switched linear systems with multi-equilibrium points (PSLS-MEP).
First, a sufficient condition is presented for the regional stability of PSLS-MEP via a common linear Lyapunov function. Second,
by establishing multiple Lyapunov functions, a dwell time based condition is proposed for the regional stability analysis. Third,
a suprasphere which contains all equilibrium points is constructed as a stability region of the considered PSLS-MEP, which is less
conservative than existing results. Finally, the study of an illustrative example shows that the obtained results are effective in the
regional stability analysis of PSLS-MEP.
Keywords: Regional stability, positive switched linear system, multi-equilibrium points, common linear Lyapunov function, multiple
Lyapunov function.
1
Introduction
Switched systems whose subsystems operate according to
a switching law have found wide applications in control systems, power systems, neural networks, process control, traffic control and many other fields[1−3] . The stability analysis
of switched systems has attracted a great deal of attention
in the last three decades[4−11] . Shorten and Narendra[8]
proposed a necessary and sufficient condition for the existence of common Lyapunov function. In addition, multiple Lyapunov function method[10] and average dwell time
technique[11] are viewed as two valid ways to study the stability for switched systems. However, most of the existing
results[12−16] typically focused on the stability of switched
systems with a common equilibrium point.
During lots of switching events, the equilibrium point
might change because of external disturbances. For example, in power systems, cutting machine and load cutting
control could change the structure of the power network,
meanwhile the operating point (i.e., equilibrium point) of
the corresponding system may also change. In mechanical
systems, mechanical vibration will lead to the position deviation of the equilibrium. In addition, the phenomenon
of multi-equilibrium points (MEP) is found widely in the
population dynamic system, network control system, missile launching system, unmanned aerial vehicle flight control, etc.
Therefore, the model of switched systems with MEP can
better describe many natural phenomena[17−19] . In referResearch Article
Manuscript received April 16, 2015; accepted September 8, 2015;
published online June 20, 2016
This work was supported by National Natural Science Foundation
of China (No. 61374065), and the Research Fund for the Taishan
Scholar Project of Shandong Province.
Recommended by Associate Editor Qing-Long Han
c Institute of Automation, Chinese Academy of Sciences and
Springer-Verlag Berlin Heidelberg 2016
ences [17, 18], by using the method of quadratic Lyapunovlike function, the authors provided a rough estimation of
stability region. The stability properties for a kind of nonlinear switched systems with MEP were studied in [19], and
the trajectory is demonstrated to globally converge to a
connected superset. However, it should be pointed out that
the boundaries of stability region given by all mentioned
methods are relatively conservative.
On the other hand, a positive system whose states, inputs
and outputs only take nonnegative values has been studied
widely in the last two decades[20−28] . A finite number of
positive linear systems running under some switching law is
called a positive switched linear system (PSLS), which has
numerous applications in engineering, economics, social science, management science, biology, medicine, etc. During
the past two or three decades, the stability theory of PSLSs
with a common equilibrium point has been well studied, and
a number of effective methods have been proposed. Among
them, the copositive Lyapunov function[24, 26] is specifically
constructed for the PSLSs and is proved to have more excellent characteristics.
In this paper we study the regional stability of PSLSMEP by using the Lyapunov function method, and present
several new results. The main contributions of this paper
are as follows. 1) Both common linear Lyapunov function
method and multiple Lyapunov function method are proposed for the regional stability of PSLS-MEP. It is noted
that one can hardly construct a linear Lyapunov function
for the regional stability of general switched systems with
MEP. This is the main difference between our results and
the existing ones[17−19] . 2) Using our methods, it is very
convenient to obtain that the trajectory of PSLS-MEP enters and remains in a suprasphere which contains all equilibrium points. Compared with [17−19], the stability region
given in our work is much less conservative (please see Re-
214
International Journal of Automation and Computing 14(2), April 2017
mark 3.5 and practical example below).
The remainder of the paper is organized as follows. Some
necessary preliminaries are introduced in Section 2. The
regional stability analysis for PSLS-MEP are carried out in
Section 3. An illustrative example is given in Section 4 to
show the effectiveness of our main results, which is followed
by conclusions in Section 5.
Notations. The notations of this paper are fairly stann×n
denote the set of all
dard. R (R+ ), Rn (Rn
+ ) and R
real (nonnegative) numbers, the set of n-dimensional (nonnegative) vectors, and the space of n × n-dimensional real
matrices, respectively. A 0 ( 0, 0, ≺ 0) denotes that
all elements of matrix A are nonnegative (non-positive, positive, negative). AT stands for the transpose of matrix A.
Mn denotes the set of n × n Metzler matrix (A real matrix
is called a Metzler matrix if its off-diagonal elements are
nonnegative). ·
means the Euclidean norm of vector in
Rn , i.e., x = |x1 |2 + |x2 |2 + · · · + |xn |2 . For any two
points x1 ∈ Rn and x2 ∈ Rn , the distance between x1 and
x1 − x2 . “sgn” denotes the
x2 is denoted by d(x1 , x2 ) =⎧
⎪
if x >0
⎨ 1,
sign function, i.e., sgn(x) =
0,
if x = 0
⎪
⎩
−1, if x < 0.
2
Aσ xeσ 0, ∀ σ ∈ l.
Proof. We first consider the following system:
ẋ(t) = A(x(t) − xe )
x(t0 ) = x0 .
Apparently, the trajectory of system (2) can be described
by
t
− Axe eA(t−τ ) dτ.
x(t) = eA(t−t0 ) x0 +
t0
From Lemma 1, we can conclude that for any initial state
x(t0 ) 0, the corresponding trajectory x(t) 0 is equivalent to A ∈ Mn , and Axe 0. This together with Lemma
4 in [13] shows that the conclusion holds.
Remark 1. It should be point out that the system matrices satisfying Aσ ∈ Mn and Aσ xeσ 0 (xeσ 0) are
widespread.
For
example, there is a 2-dimensionalmatrix
a b
e
e
A =
and an equilibrium point x =
,
c d
f
where a, b, c, d, e, f are constants. A simple calculation
shows that when a, b, c, d, e, f satisfy
⎧
b≥0
⎪
⎪
⎪
⎪
⎪
c≥0
⎪
⎪
⎪
⎨ e≥0
⎪
f≥0
⎪
⎪
⎪
⎪
⎪
ae + bf ≤ 0
⎪
⎪
⎩
ce + df ≤ 0
Preliminaries
Consider the following PSLS-MEP:
ẋ(t) = Aσ(t) x(t) − xeσ(t)
(1)
x(t0 ) = x0
where x(t) : R+ → Rn
+ is the state vector, the switching
signal σ(t) : [t0 , +∞) → l = {1, 2, · · · , l} is a piecewise
constant function, l is the number of subsystems, Aσ (σ ∈ l)
are the system matrices, xeσ = (xeσ1 , · · · , xeσn )T (σ ∈ l) are
the equilibrium points, and the initial state x0 0. The
switching moment sequence is denoted by {tk }+∞
k=0 , which
satisfies t0 < t1 < t2 < · · · < tk < · · · , where t0 (≥ 0)
denotes the initial moment, tk denotes the k-th switching
instant, τk = tk − tk−1 (k = 1, 2, · · · ) are the dwell time of
the k-th activated subsystem, and τk > 0. Throughout this
paper, we suppose that: 1) There are finite switching times
in any bounded time interval. 2) Every subsystem can be
activated in our considered operating intervals. 3) All the
subsystems are asymptotically stable.
Next, we give some preliminary results on positive systems.
Definition 1[12] . The system (1) is said to be positive if for any initial condition x(t0 ) 0 and any switching signal σ(t), the corresponding trajectory x(t) satisfies
x(t) 0, ∀ t ≥ t0 .
Lemma 1[12] . Let A ∈ Rn×n . Then, eAt 0, ∀ t ≥ 0,
if and only if A ∈ Mn .
We present a necessary and sufficient condition for the
positivity of the system (1).
Lemma 2. For any x(t0 ) 0 and any switching signal
σ(t), system (1) is positive if and only if Aσ ∈ Mn and
(2)
xe 0, A ∈ Mn and Axe 0 hold.
Finally, we give the concept of regional stability for system (1).
Definition 2[17] . System (1) is said to be regionally
stable under arbitrary switching signal σ(t), if there exists
a nonempty bounded set N ⊆ Rn
+ , such that for any x(t0 ) 0, lim x(t) ∈ N . The set N is called a stable region of
t→+∞
system (1).
Remark 2. Although all the subsystems are asymptotically stable, system (1) cannot converge to a point under
arbitrary switching signals because all the subsystems share
different equilibrium points. In fact, the trajectory of (1)
will enter into a region which contains all the equilibrium
points and some trajectories.
3
Main results
In this section, we study the regional stability of the system (1), and present the main results of this paper. To
this end, we give the following two propositions, which are
crucial in the proof of our main results.
Given ai ∈ Rn (i ∈ n = {1, 2, · · · , n}), denote by
a =
n
1
ai
n i=1
215
Z. Liu and Y. Z. Wang / Regional Stability of Positive Switched Linear Systems with Multi-equilibrium Points
where
and
a) ≤ max d(
a, ai )}.
M = {x ∈ Rn | d(x, D+ |xi (t)| = lim
i∈n
h→0+
Proposition 1. If there exists a point a ∈ Rn such that
n
i=1
n
d(
a, ai ) = minn
x∈R
d(x, ai )
denotes the right derivative of |xi (t)|, i ∈ n, and Aσ =
(aij )(σ) , σ ∈ l.
Proof. 1) When xi (t) = xeσi , i.e., xi (t) − xeσi = 0, we get
i=1
D+ |xi (t) − xeσi | = |D+ xi (t)| =
then a ∈ M.
/ M , let a be the intersection of the
Proof. For any a∗ ∈
∗
line segment aa and ∂M (the boundary of the suprasphere
M ). For a special case, the three points ai (i ∈ n), a , a∗ are
collinear, and thus one immediately finds that the length of
the line segments satisfy a∗ ai > a ai , namely
|
(σ)
aii |(xi (t) − xeσi )| =
n
2) When xi (t) = xeσi , by the fact that
(σ)
aii ≤ 0,
i=1
D+ (xi (t) − xeσi ) sgn(xi (t) − xeσi ) =
n
n
i=1
x∈R
(σ)
aii (xi (t) − xeσi ) sgn(xi (t) − xeσi ) =
n
d(x, ai ) =
i=1
(σ)
aii |(xi (t) − xeσi )| ≤
n
n
i=1
if x ∈ M . For any a∗ ∈
/ M , we are always able to find a
point a ∈ ∂M ⊆ M such that
d(a , ai ) <
i=1
j=1
d(a∗ , ai ).
For the sake of convenience, we introduce symbols
. 1 e
xσ 0
x̄ =
l σ∈l
and
.
r = max{d(x̄, xeσ )} ≥ 0.
σ∈l
(4)
The suprasphere is defined as
N = {x ∈ Rn
+ | d(x, x̄) ≤ r}.
(5)
i=1
Proposition 2. Assume that x(t) is a solution of system
(1). Then,
D+ |xi (t) − xeσi | ≤
(σ)
aij |xj (t) − xeσj |.
d(
a, ai ).
i=1
n
(σ)
aij (xj (t) − xeσj ) sgn(xi (t) − xeσi )+
j=1,j=i
d(a , ai ).
Therefore, a ∈ M.
Remark 3. From Proposition 1, we know that for any
n
x ∈ Rn ,
d(x, ai ) reaches the minimum value if and only
n
(σ)
aij (xj (t) − xeσj ) sgn(xi (t) − xeσi )+
j=1,j=i
Hence, for any a ∈
/ M , we are always able to find a point
a ∈ ∂M ⊆ M such that
n
(σ)
aij (xj (t) − xeσj ) sgn(xi (t) − xeσi ) =
j=1
i=1
d(a∗ , ai ) > minn
i = j
D+ |xi (t) − xeσi | =
∗
n
(σ)
aij ≥ 0,
we have
holds for any i ∈ {1, 2, · · · , n}. That is to say
n
(σ)
aij |xj (t) − xeσj |.
j=1
d(a∗ , ai ) >
(σ)
aij |xj (t) − xeσj |+
j=1,j=i
d(a , ai ) > d(a , ai )
n
(σ)
aij (xj (t) − xeσj )| ≤
n
∗
If the three points ai (i ∈ n), a , a are not collinear, we
know that the three points a∗ , a , ai (i ∈ n) can be used
as three vertices in a triangle. Next, we will explain all the
triangles are obtuse triangles, and the line segments a∗ ai
are the edges with respect to obtuse angles in the triangle
a∗ a ai . First, one can construct a tangent plane of M
through the point a . Let a∗i be the intersection of the
tangent plane and the line segment ai a∗ . It is easy to see
that the points ai , a , a∗i and a∗ are in the same plane,
and the angle ∠a∗ a a∗i is a right angle. Then, the angle
∠ai a a∗ is an obtuse angle in the triangle a∗ a ai . From
the relationship between side and angle, one can conclude
that the size of the line segment a∗ ai is greater than the
line segment a ai , i.e.,
∗
n
j=1
d(a∗ , ai ) > d(a , ai ).
|xi (t + h)| − |xi (t)|
h
n
j=1
(σ)
aij |xj (t) − xeσj |
(3)
In the following, we establish sufficient conditions for the
regional stability of system (1) by two different methods.
Theorem 1. System (1) is regionally stable if there
exists a vector ξ = (ξ1 , ξ2 , · · · , ξn )T ∈ Rn
+ satisfying
ξ T Aσ 0, ∀ σ ∈ l.
(6)
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International Journal of Automation and Computing 14(2), April 2017
Proof. For any t > t0 , assume that t ∈ [tk , tk+1 ) holds
for some k ≥ 0. We establish the candidate common linear
Lyapunov function as
l
V (t) =
ξi |xi (t) − xeσi |, t ≥ t0 .
(7)
Calculate the right derivative
D+ V = lim
h→0+
V (t + h) − V (t)
h
along the trajectories of system (1) under an arbitrary
switching signal σ(t). It is easy to see from Proposition 2
that
l n
σ=1 i=1
ξi
n
(σ)
aij |xj (t) − xeσj | =
where
Gσ (t) =
(|x1 (t) − xeσ1 |, |x2 (t) − xeσ2 |, · · · , |xn (t) − xeσn |)T 0.
This together with (6) implies that
(8)
In addition, V (t) is a positive function and has the minil
d(x, xeσ ) reaches the minimum value.
mum value Vmin if
σ=1
From the monotone bounded theorem, we know that
lim V (t) = Vmin .
t→+∞
σ=1
d(x, xeσ ) will reach the minimum value
as the time t → +∞. By Proposition 1, it yields that x(t)
enters N after some moment.
In the following part, we will explain the trajectory x(t)
not only enters the suprasphere N but also remains in it.
Might as well it can be supposed that the moment T is the
first time that trajectory x(t) enters N , i.e.,
d(x(T ), x̄) = r,
T > 0.
Assume, for the sake of contradiction, that for any > 0,
/ N , i.e.,
there exists t∗ = T + such that x(t∗ ) ∈
d(x(t∗), x̄) > r,
t∗ > T.
Due to the arbitrariness of , the three points x(t∗ ), x(T ), x̄
are approximated to be on the same line. From the Propo/ N , one can claim
sition 1 and Remark 2, for any x(t∗ ) ∈
that
l
d(x(t∗ ), xeσ ) >
σ=1
l
d(x(T ), xeσ ).
σ=1
On the other hand, together with (8), we have
V (t∗ ) ≤ V (T ),
for
t∗ > T
d(x(T ), xeσ ).
(10)
σ=1
Apparently, (9) and (10) are contradictory. That is to
say, the trajectory x(t) will not escape from N , once it
enters N . Hence, for any initial condition x(t0 ) 0 and
any switching laws, system (1) is regionally stable, and N
is the stability region.
Remark 4.
Based on the method of quadratic
Lyapunov-like function, Guo and Wang[17, 18] provided the
estimation of stability region as
Ω = {x ∈ Rn : (x − x̄)T P (x − x̄) ≤ Cp }
σ∈L
ξ T Aσ Gσ (t)
D+ V ≤ 0.
l
By a simple calculation, we have
λmax (P )
max Aσ (x̄ − xeσ ).
x − x̄ ≤ 2
λmin (P ) σ∈L
σ=1
l
d(x(t∗ ), xeσ ) ≤
where Cp = 4λμ2 , λ = λmax (P ), μ = max Aσ (x̄ − xeσ ).
j=1
l
In other words,
l
σ=1
n
σ=1 i=1
D+ V ≤
i.e.,
(9)
(11)
Comparing (11) with (5), it is obvious to see that
λmax (P )
max{Aσ × x̄ − xeσ }.
r≤2
λmin (P ) σ∈L
Therefore, our estimation is better than [17, 18] .
Remark 5. From the above analysis, one can see that
the suprasphere N is closely connected to the Lyapunov
function. It is noted that how to find the accurate boundary
of stability region for a PSLS-MEP is a very challenging
problem.
When the number of subsystems is large, it is difficult to
find a common ξ satisfying ξ T Aσ 0. In view of this kind
of situation, we can weaken the conditions in Theorem 1
and obtain the following theorem.
Theorem 2. Suppose that there exist l constant vectors
(σ)
(σ)
(σ)
ξ (σ) = (ξ1 , ξ2 , · · · , ξn )T ∈ Rn
+ (σ ∈ l) and constants
μ > 1, α > 0 such that the following conditions hold:
1) (ξ (ik ) )T Ak + α(ξ (ik ) )T 0, k ∈ l, t ∈ [tk , tk+1 ), σ =
ik ∈ l.
2) (ξ (i) )T ≤ μ(ξ (j) )T , i, j ∈ l.
ln μ
3) The average dwell time τa ≥
.
α
Then, system (1) is regionally stable under switching signals σ(t), and N defined in (4) is an estimation of stability
region.
Proof. For any t > 0, assume that t ∈ [tk , tk+1 ) for some
k ≥ 0 and σ(t) = ik ∈ l. Denote N = N (t, t0 ) by the number of switching times on the interval [t0 , t], and for each
subsystem, we establish the multiple Lyapunov function as
Vσ (t) = Vik (t) =
n
l ik =1 i=1
(ik )
ξi
|xi (t) − xeik i |
for t ≥ t0 and σ(t) = ik .
By calculating the right derivative, we get
D+ Vik ≤
n
l ik =1 i=1
(ik )
ξi
n
j=1
(i )
aijk |xj (t) − xeik j |
(12)
Z. Liu and Y. Z. Wang / Regional Stability of Positive Switched Linear Systems with Multi-equilibrium Points
and
D+ Vik + αVik ≤
n
l (ik )
ξi
ik =1 i=1
α
j=1
n
l (ik )
ξi
ik =1 i=1
l
n
(i )
aijk |xj (t) − xeik j |+
|xi (t) − xeik i | =
(ξ (σ) )T Aσ Gσ (t)+
σ=1
α
l
(ξ (σ) )T Gσ (t) =
σ=1
l
217
Generally speaking, a data communication network consists of four parts: control center, communication terminal,
communication links and data-interchange nodes. Starting from the control center, different data packets are sent
to the communication terminal through the corresponding
communication links and data-interchange nodes. If the
amount of data transmitted over the network is too large,
then the network might occur, the phenomenon of congestion and the data packets might get lost in the process of
transmission. On the contrary, if the amount of data transmitted over the network is too small, it is a huge waste
of network resources. In order to achieve a win-win situation, every subnetwork should have a relatively appropriate
amount of data in practice.
[(ξ (σ) )T Aσ + α(ξ (σ) )T ]Gσ (t).
σ=1
According to the condition 1), we derive that
D+ Vσ + αVσ ≤ 0
for t ∈ [tk , tk+1 ). Hence,
Vσ (t) ≤ e−α(t−tk ) Vσ (t+
k ).
By the conditions 2) and 3), we have
ViN (t) ≤ e−α(t−tN ) ViN (t+
N) ≤
Fig. 1
μe−α(t−tN ) ViN −1 (t−
N) ≤ · · · ≤
μN e−α(t−t0 ) Vi0 (t0 ).
Note that the switching times and the average dwell time
satisfy
N ≤ N0 +
t − t0
τa
which can be reduced to
ViN (t) ≤ μN0 e−(α−
ln μ
)(t−t0 )
τa
Vi0 (t0 ).
For a small positive real number , the Lyapunov function
l
V (t) → as the time t → +∞. That is to say,
d(x, xeσ )
σ=1
will reach the minimum value as the time t → +∞. By the
similar analysis method as Theorem 1, it yields that x(t)
enters and remains in N .
Remark 6. It is noted that the common linear Lyapunov function method and the multiple Lyapunov function
method have their own advantages, and cannot contain each
other. On one hand, the condition (ξ (ik ) )T Ak + α(ξ (ik ) )T 0 in Theorem 2 is weaker than the condition in Theorem 1.
On the other hand, Theorem 1 is applicable to arbitrary
switching signals, while Theorem 2 is based on the average
dwell time switching signals.
4
Numerical example
In this section, we apply the obtained results to the analysis of data communication networks.
Data communication networks with three nodes
According to [14−16], the data communication network
showed in Fig. 1 can be described by the following PSLSMEP
ẋ(t) = Aσ x(t) − xeσ
(13)
x(t0 ) = x0
where x(t) ∈ R3+ denotes the amount of data transmitted
over the network, xeσ is the known appropriate amount of
data, σ = {1, 2}, and
⎛
⎞
−0.6 0.02 0.03
⎜
⎟
A1 = ⎝ 0.03 −0.4 0.04 ⎠
0.02 0.03 −0.5
⎛
⎞
−0.3 0.04 0.04
⎜
⎟
A2 = ⎝ 0.02 −0.5 0.03 ⎠ .
0.05 0.06 −0.4
Note that σ = 1 denotes the idle-time network, while σ = 2
denotes the busy-time network. Without loss of generality,
we suppose that
1) The maximum amount of data that the idle-time
(busy-time) network can bear is 2 Gb (4 Gb). That is to
say, x1 (t) + x2 (t) + x3 (t) ≤ 2 (x1 (t) + x2 (t) + x3 (t) ≤ 4).
2) The appropriate amount of data for idle-time model
is xe1 = (0.6, 0.6, 0.7)T , and for busy-time model is xe2 =
(1.4, 1.2, 1.3)T .
3) Take the initial moment t0 = 0, and the initial amount
of data that needs to be sent across the network as x(t0 ) =
(0.45, 0.9, 1.5)T .
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International Journal of Automation and Computing 14(2), April 2017
It is easy to verify that Aσ xeσ 0 (σ ∈ {1, 2}) and both
subsystems are asymptotically stable (see Fig. 2).
we get the stability region (provided in [17, 18]) Ω = {x :
x(t) − x̄ ≤ 0.708 2}. Obviously, N is smaller than Ω;
2) From reference [19], N (i) (k) = {x ∈ R3+ : x − xei ≤
N (i) (k), α(i) (k) =
max x1 − x2 ,
k}, N (k) =
x1 ,x2 ∈N(k)
i=1,2
M (i) (k) = {x ∈ R3+ : x − xei ≤ α(i) (k)}, and the stability
region is
L(k) =
M (i) (k).
i=1,2
Fig. 2
Stability of each subsystem
Choose k = 0.02.
In order to make a more intuitive comparison between
N and L(k), we provide the figures of stability regions in
different coordinate planes, please see Figs. 4 − 6, where the
region bounded by solid line is N , and the region bounded
by dashed line is L(k). Clearly, our result is less conservative than [19].
At the same time, by a direct calculation, we obtain
x̄ = (1, 0.9, 1)T
d(x̄, xe1 ) = d(x̄, xe2 ) = 0.583 1
and the stable region
N = {x ∈ R3+ |d(x, x̄) ≤ 0.583 1}
is a sphere in R3+ (see Fig. 3). Choose ξ = (1, 1, 1)T such
that the condition in Theorem 1, i.e., ξ T Aσ 0 holds. In
our simulation, σ is a periodic switching signal and the dwell
time is randomly generated.
Fig. 3 tells us that for a given initial amount of data x0 0, the trajectory of system (13) enters and remains in N at
last. Our simulation result is consistent with the theoretical
result.
Fig. 3
Fig. 4
Stability regions in x1 -x2 plan
Fig. 5
Stability regions in x1 -x3 plane
Stability region and trajectory of the system (13)
In order to better illustrate the effectiveness of the proposed method, we list the stability regions given in [17−19],
respectively.
1) By a simple calculation, we have A1 = 0.607 3,
A2 = 0.518 4. Choose P = I3 . Together with (11),
Z. Liu and Y. Z. Wang / Regional Stability of Positive Switched Linear Systems with Multi-equilibrium Points
219
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Fig. 6
5
Stability regions in x2 -x3 plane
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Conclusions
In this paper, we have studied the regional stability of
positive switched linear systems with multiple equilibrium
points. The innovation of our work is that both common
linear Lyapunov function and multiple Lyapunov functions
are constructed to solve the regional stability of PSLS-MEP.
Moreover, an effective estimation of stable region has been
given by using our method. A numerical example has been
given to illustrate our results.
Compared with [17−19], the stability region given in our
work is much less conservative. Future works will study
how to obtain the accurate boundary of stability region for
PSLS-MEP.
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Zhi Liu
received the B. Sc. degree from
School of Mathematics and Information,
Ludong University, China in 2009, and the
M. Sc. degree from School of Mathematical
Science, University of Jinan, China in 2013.
Since 2013 she is a Ph. D. degree candidate
at School of Control Science and Engineering, Shandong University, China.
Her research interests include positive
systems and switched systems.
E-mail: [email protected] (Corresponding author)
ORCID iD: 0000-0002-4745-9025
Yu-Zhen Wang graduated from Tai an
Teachers College, China in 1986, received
the M. Sc. degree from Shandong University
of Science and Technology, China in 1995,
and the Ph. D. degree from Institute of Systems Science, Chinese Academy of Sciences,
China in 2001. Since 2003, he is a professor
with School of Control Science and Engineering, Shandong University, China, and
now the dean of the School of Control Science and Engineering, Shandong University. From 2001 to 2003, he worked as a
postdoctoral fellow in Tsinghua University, China. From March
2004 to June 2004, from Februery 2006 to May 2006 and from
November 2008 to January 2009, he visited City University of
Hong Kong as a research fellow. From September 2004 to May
2005, he worked as a visiting research fellow at the National University of Singapore. He received the Prize of Guan Zhaozhi in
2002, the Prize of Huawei from the Chinese Academy of Sciences
in 2001, the Prize of Natural Science from Chinese Education
Ministry in 2005, and the National Prize of Natural Science of
China in 2008. Currently, he is an associate editor IMA Journal of Mathematical Control and Information, and a Technical
Committee member of IFAC.
His research interests include nonlinear control systems,
Hamiltonian systems and Boolean networks.
E-mail: [email protected]