Practical stability of linear time-varying delay systems

Ha Noi Metroplolitan University

12

PRACTICAL STABILITY OF LINEAR TIMETIME-VARYING
DELAY SYSTEMS
Le Van Hien
Hanoi National University of Education
Abstract: This paper is concerned with the problem of practical stability of linear timevarying delay systems in the presence of bounded disturbances. Based on some
comparison techniques associated with positive systems, explicit delay-independent
conditions are derived for determining a neighborhood of the origin which ultimately
bounds all state trajectories of the system.
Keywords: Practical stability, time-varying delay, Metzler matrix.
Email:
Received 29 July 2018
Accepted for publication 15 October 2018

1. INTRODUCTION
In practical systems, there usually exists an interval of time between a stimulation and
the system response [1]. This interval of time is often known as time delay of a system.
Since time-delay unavoidably occurs in engineering systems and usually is a source of
poor performance, oscillations or instability [2], the problem of stability analysis and
control of time-delay systems is essential and of great importance for theoretical and
practical reasons [3]. This problem has attracted considerable attention from the
mathematics and control communities, see, for example, [4−10].
When considering long-time behavior of a system, the framework of Lyapunov
stability theory and its extensions for time-delay systems, the Lyapunov-Krasovskii and
Lyapunov-Razumikhin methods, have been extensively developed [3]. However, realistic
systems usually exhibit characteristics for which theoretical definitions in the sense of
Lyapunov can be quite restrictive [11]. Namely, the desired state of a system may be
mathematically unstable in the sense of Lyapunov, but the response of the system oscillates

close enough to this state for its performance to be considered as acceptable. Furthermore,
in many control problems, especially for systems that may lack an equilibrium point due to
the presence of disturbances or constrained states, the aim is to bring those states close to


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certain sets rather than to a particular state [12-16]. In such situations, the concept of
practical stability is more suitable and meaningful. Practical stability, also referred to as
ultimate boundedness with a fixed bound [17], was first proposed in [18], retaken and
systematically introduced in [19] to address some potential practical limitations of
Lyapunov stability. These stability notions not only provide information on the stability of
the system, but also characterize its transient behavior with estimations of the bounds on
the system trajectories. During last decades, considerable research attention has been
devoted to study the practical stability of dynamical systems. To mention a few, we refer
the reader to recent papers [11, 14-16, 20-26] and the references therein.
It is worth to mention here that, in most of the existing results, the framework of
Lyapunov and its variants, have been suitably developed as the main approach to derive
conditions for some specific types of practical stability. Particularly, in [21] some results
parallel to the Lyapunov results have been proposed for the strict practical stability of a
general class of delay differential systems. Using the idea of perturbing Lyapunov
functions combining with the comparison principle, the authors in [23] established
sufficient conditions for various types of strict practical stability of nonlinear impulsive
delay-free systems. In [15] some results on practical stability of nonlinear delay-free
switched systems without a common equilibrium and under a time-dependent switching
signal were given by employing the idea of direct method proposed in [22]. Some
interesting applications of practical stability to realistic systems were investigated in [24]
for brake model of a bike and in [25] for a congestion control model in computer networks

by using some Lyapunov-like functions. In [14] and [16], practical stabilization and
extended Lyapunov methodology was developed for some classes of nonlinear control
systems where the measurement is sampled and possibly delayed. In [11], based on the
Lyapunov-Krasovskii functional (LKF) proposed in [4], the authors derived sufficient
practical stability and stabilizability conditions for LTI systems with constant delays in
terms of feasible linear matrix inequalities (LMIs). As discussed by the authors, the
approach allows one to constructively obtain bounds on the practical stability region. By
extending this approach to the case of neutral systems, the authors in [26] proposed a set of
bilinear matrix inequalities (BMIs) for practical exponential convergence of a class of
nonlinear neutral systems with multiple constant delays and bounded disturbances.
Although practical stability provides a more relaxed concept of stability, there are very
few studies especially for time-varying delay systems. Furthermore, when dealing with
time-varying systems, the developed methodologies such as LKF and its variants either
lead to matrix Riccati differential equations (RDEs) [27] or indefinite LMIs. So far, there


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has been no efficient computational tool available to solve RDEs or indefinite LMIs. In
addition, the constructive approaches proposed in the aforementioned works are
inapplicable to time-varying systems. Therefore, an alternative and efficient approach to
address the problem of practical stability of time-varying systems with delays is obviously
necessary. This motivates us in the present research.
In this paper, we address the problem of practical stability of linear time-varying
systems with time-varying delay and bounded disturbances. We present a constructive
approach based on some techniques developed for positive systems which we have
successfully applied to linear time-varying systems with delays [28, 29]. New explicit
delay-independent conditions are derived for determining a neighborhood of the origin

which attracts exponentially all state trajectories of the system. In addition, our conditions
also guarantee the Lyapunov exponential stability of the system in the absence of input
disturbances.
The remainder of this paper is organized as follows. In Section 2, we present the
problem statement, review some background results and introduce some notations that will
be used throughout this paper. The main results are presented in Section 3. Illustrative
examples and a conclusion are given in Section 4 and Section 5, respectively.

2. PRELIMINARIES
Notation. ℝ and ℕ denote the set of real numbers and natural numbers, respectively.
For a given n ∈ ℕ , n≜{1,2,…, n}. ℝ n is the n-dimensional vector space endowed with the
norm ‖x‖

= max

|xi | for x = (xi) ∈ ℝ n . The non-negative orthant of ℝ n will be

∈

denoted by ℝ n+ . By int(X), we denote the interior of the subset X ⊂ ℝ n . Let ℝ m×n be the set
of all m × n real matrices. For a matrix A ∈ ℝ m×n , ri ( A) ∈ ℝ1×m , denotes the ith row of A.
Inequalities between vectors will be understood componentwise. Specifically, for u = (ui)
and

= ( i) in ℝ n , u ≥

write

u

instead

≫

means
of

int ( ℝ n+ ) = { x ∈ ℝ n : x ≫ 0} . Denote

for all ∈

≥
> .
min

In

and if

particular,

= min ∈

i

=

if and only if > 0, otherwise

for all ∈

then we

ℝ = {x ∈ ℝ : x ≥ 0}
n
+

then

v = (vi ) ∈ int ( ℝ n+ ) . We also specifically use the notation

, that means

>

n

and

> 0 for any vector

= max{ , 0} for real number

= 0.

Consider the following linear time-varying system with delay


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xɺ (t ) = A(t ) x (t ) + B (t ) x(t − τ (t )) + d (t ), t ≥ 0,
x(t ) = φ (t ), t ∈ [−τ max , 0],

=

where

! ∈ ℝ n is the system state vector and "

(1)

= "

! ∈ ℝ n is

unknown input disturbance vector, A(t ) = (aij (t )) ∈ ℝ n×n and B(t ) = (bij (t )) ∈ ℝ n×n are timevarying system matrices whose elements are assumed to be continuous on ℝ + , τ (t ) is a
time-varying delay and φ (.) ∈ C ([−τ max , 0], ℝ n ) is the vector-valued initial function
specifying

the

initial

| φi | = sup | φi (t ) | and φ
−τ max ≤t ≤ 0

state
∞

of

the

system, φ (t ) = (φi (t )) ∈ ℝ n .

Let

us

denote

= max i∈n | φi | .

Remark 2.1. In this paper, the time delay #
is assumed to be continuous in time,
≤ # %& , for all ≥ 0, where the
not necessarily differentiable, and satisfies 0 ≤ #
upper bound # %& is a known constant. We do not impose any restriction on the rate of
(such as slowly time-varying condition #(
≤ #) <1 for all ≥ 0 . This
change of #
means that our derived conditions can be applicable to systems with fast time-varying
delay # .
First, we introduce the following definition.

Definition 2.1. For a given positive number * , system (1) is said to be μ-practically
stable if for any φ (.) ∈ C ([−τ max , 0], ℝ n ) these exists a transient time T = T( µ , φ ) ≥ 0 such
that x(t ,φ )

∞

≤ µ for all

≥ ,.

Our aim in this paper is to derive explicit conditions for determining μ-neighborhood
and finite transient time T guaranteeing the practical stability of system (1). By using a
novel approach, we propose new delay-independent conditions via spectral properties of
Metzler matrices ensuring practical exponential convergence of all state trajectories of
the system.
For the rest of this section, we review some basic background that will be used in the
next Section. At first we recall here some properties of Metzler matrix (see, [29, 30] for
more details). A matrix M ∈ ℝ n×n is said to be Metzler matrix if all off-diagonal elements of
M are nonnegative, i.e., if M = (aij ) ∈ ℝ n×n is a Metzler matrix then - . ≥ 0 for all ≠ 0.
For a matrix M ∈ℝn×n , the spectrum of M is defined as σ ( M ) = {λ ∈ ℂ : det(λ I n − M ) = 0}
and the spectral abscissa of M is given by µ ( M ) = max{Reλ : λ ∈ σ ( M )} . We now
summarize some properties of Metzler matrices in the following proposition.


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Proposition 2.1. Let M ∈ ℝ n×n be a Metzler matrix. The following statements are
equivalent
(i)

µ ( M ) < 0.

(ii) M is invertible and 123 ≤ 0.
(iii) There exists a vector ξ ∈ int (ℝ n+ ) such that M ξ ≪ 0 .
(iv) For any b ∈ int (ℝ n+ ) , there exists x ∈ ℝ n+ such that Mx + b = 0.
(v) There exists a vectorη ∈ int (ℝ n+ ) such that M Tη ≪ 0 .
(vi) For any x ∈ ℝ n+ \ {0}, the row vector

4

1 has at least one negative entry.

From Proposition (1), we obtain the following result.

Proposition 2.2. Let M ∈ ℝ n×n be a Metzler matrix satisfying one of the equivalent
conditions (i)-(iv) in Proposition 2.1. Then, there exists a vector ξ ∈ int (ℝ n+ ) , ξ

∞

= 1 , such

that M ξ ≪ 0 .

‖ of the state, we use the notion of upper-right
In order to estimate the norm ‖
Dini derivative of continuous real-valued function. Let v (.) : ℝ → ℝ be a continuous
function. The upper-right Dini derivative of v (.) , denoted by D + v(.) , is defined as follows

D + v(t ) = lim sup
h →0

+

v(t + h) − v(t )
, t ∈ ℝ.
h

3. MAIN RESULTS
The following assumptions will be used in the derivation of our results.
(A1) The system matrices A(t ) = (aij (t )) ∈ ℝ n×n and B(t ) = (bij (t )) ∈ ℝ n×n satisfy the
following conditions

-

≤ -5 , 6- .

(A2) The disturbance vector "
positive constant γ such that

|"

6 ≤ -5 , ≠ 0, 67 .
= "

| ≤ γ , ∀ ≥ 0,

∈

6 ≤ 75 . , ∀ ≥ 0, , 0 ∈

.

is bounded, that means, there exists a

.

Remark 3.1. For any initial function φ (.) ∈ C ([−τ max , 0], ℝ n ) , there exists a unique
solution x (t , φ ) of (1) defining on [ −τ max , ∞ ] [1]. On the other hand, according to (A2),


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system (1) may not have an equilibrium point. Particularly, = 0 is neither an equilibrium
point of (1) due to not vanished disturbance nor a necessarily stable motion.
We denote A = ( aij ) , B = (bij ) and M = A + B . Clearly, M is a Metzler matrix.
Therefore, if M satisfies one of the equivalent conditions in Proposition 2.1 then, by
Proposition 2.2, there exists a vector ξ ∈ int ( ℝ n+ ) , ξ

∞

= 1 ,such that M ξ ≪ 0 .Now, we are

in a position to present our main result in the following theorem.
Theorem 3.1. Let assumptions (A1)–(A2) hold and assume that the matrix M
satisfies one of the equivalent conditions (i)-(vi) in Proposition 2.1. Then, system (1) is
γ
*-practically stable for any * > ∗ . The transient time T ( µ , φ ), φ (.) ∈ C ([−τ max , 0], ℝ n ) , is
given by


 φ∞ γ
− *


1
ξ
m
min
 ln 
T ( µ ,φ ) =  σ  µ − γ


m*


0








m*

ξ min

∞

> µξ min

elsewhere .

Here ξ ∈ int ( R n+ ) is a vector satisfying ξ

δ* =

if φ

∞

= 1 and M ξ ≪ 0 , m* = ( −Mξ )min ,

and σ = min i∈n σ i , where σ i is the unique positive solution of the scalar

equation
n

σξi + ∑ bijξ j ( eστ
j =1

max

)

− 1 − m* = 0, i ∈ n.

Moreover, any solution x (t ,φ ) of (1) satisfies the following bound

x(t , φ )

∞

≤


+κ*  φ
*
m


γ

+

−
∞

γ  −σ t
e , t ≥ 0,
δ * 

where κ * = 1 / ξ min .
Proof . We divide the proof into several steps.
Step 1. By Proposition 2.2, there exists a vector ξ ∈ int (ℝ n+ ) , ξ

∞

= 1 , such that

M ξ ≪ 0 . For convenience, we denote D = diag{aii } and AD = A − D . Then, we have

aiiξi + ri (AD + B )ξ < 0, i ∈ n.

(2)


Ha Noi Metroplolitan University

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Note that

m* = ( −Mξ )min = min i∈n {−aiiξi − ri (AD + B )ξ } > 0
and from (2) we have
aiiξ i + ri (AD + B )ξ ≤ − m* , i ∈ n.

γ
γ
, then x(t, φ ) ∞ ≤ * , for all ≥ 0. In the
*
m
m
to denote the solution x (t , φ ) if it does not make any confusion.

Step 2. We will prove that, if φ

following, we will use

(3)

∞

≤

γ
γ
, then we have xi (t ) ≤ φi ≤ ξi * , for all t ∈ [−τ max ;0], i ∈ n. For any
*
m
m
γ
given < > 1, assume that there exists an index i ∈ n and t > 0 such that xi (t ) = qξi * and
m
γ
x j (t ) ≤ qξ j * , t ∈ [0, t ], j ∈ n. Then, D + xi (t ) ≥ 0. On the other hand, it follows from
m
(1) that
Let φ

∞

≤

D + xi (t ) = sgn( xi (t )) xi (t )
n

≤ aii (t ) | xi (t ) | +

∑ | a (t ) || x (t ) |
ij

j

j =1, j ≠i
n

+ ∑ | bij (t ) || x j (t − τ (t )) | + | di (t ) |
j =1

≤ aii (t ) | xi (t ) | +

n

∑

j =1, j ≠i

n

aij (t ) | x j (t ) | +∑ bij (t ) | x j (t − τ (t )) | + γ, t ∈ [0, t ]. (4)
j =1

Thus,

qγ
(aijξi + ri ( AD + B) + γ
m*
≤ (1 − q)γ < 0

D+ xi (t ) ≤

which yields a contradiction. This shows that xi (t ) ≤ qξi
we obtain xi (t ) ≤ ξi

γ
, for all i ∈ n and t ≥ 0, and hence,
m*
xi (t ) ∞ ≤

γ
ξ
m*

∞

=

γ
,
m*

t ≥ 0.

(5)

γ
, for all t ≥ 0, i ∈ n. Let < ↓ 1
m*

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Step 3. Now, we assume φ

φi ≤ ξi

>

∞

19

γ

δ*

. Then, it is easy to verify that

γ

≤ κ*  φ
*
m


∞

−

γ 
ξi , i ∈ n.
δ * 

For each i ∈ n, consider the following scalar equation in σ ∈ [0, ∞ )
n

∑

H i (σ ) = σξi +

bijξ j (eστ max − 1) − m* = 0.

(6)

j =1, j ≠i

Since the function H i (σ ) is continuous and strictly increasing on [0, ∞), H i (0) < 0

and H i (σ ) → ∞ as σ → ∞, equation (6) has a unique positive solution σ i . In addition,
H i (σ ) < 0 for all σ ∈ (0, σ i ]. Let σ = min i∈n σ i then H i (σ ) ≤ 0 for all i ∈ n.

Let us consider the functions all vi (t ), i ∈ n, defined as follows

vi (t ) = κ *  φ


∞

−

γ  −σ t
ξ e , t ∈ [ −τ max , ∞ ).
δ *  i

(7)

For any t ≥ 0 and j ∈ n, it is clear from (7) that

vi (t − τ (t )) = κ *  φ


≤ κ*  φ


aii (t )vi (t ) +

n

∑

j =1, j ≠i

∞

∞

−

γ  −σ ( t −τ ( t ))
ξe
δ *  i

−

γ  −σ t στ
ξe e
δ *  i

max

≤ eστ max vi (t ).

n

aij v j (t ) + ∑ bij v j (t − τ (t ))
j =1

n
n


≤ β e−σ t  aiiξi + ∑ aijξ j + ∑ bijξ j eστ max 
j =1, j ≠ i
j =1


n



≤ β e −σ t  aiiξi + ri (AD + B )ξ + ∑ bijξ j eστ max − 1 
j =1



(

n


≤ β e−σ t  −m* + ∑ bijξ j eστ max − 1 
j =1



(

≤ − βσξi e−σ t , t ≥ 0, i ∈ n,

where β = κ *  φ


∞

−

γ 
.
δ * 

)

)


Ha Noi Metroplolitan University

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The above estimation leads to

vɺi (t ) > a ii vv (t )

n

n

∑

j =1, j ≠i

a ii v j (t ) + ∑ bij v j (t − τ (t )), t ≥ 0.

(8)

j =1

Next, by using the following transformation

ui (t ) = xi (t ) − ξi

γ
, t ≥ −τ max , i ∈ n,
m*

and by the same argument used in (4), we have
n

D+ui (t ) ≤ aiiui (t ) +
+

γ
m*

(a ξ

ii i

∑

n

j =1, j ≠i

aij u j (t ) + ∑ bij (t )u j (t − τ (t ))
j =1

+ ri (AD + B )ξ ) + γ
n

≤ aii ui (t ) +

∑

j =1, j ≠i

(9)

n

aij u j (t ) + ∑ bij (t )u j (t − τ (t )),
j =1

for all t ≥ 0. We now prove that ui (t ) ≤ vi (t ) for all t ≥ 0, i ∈ n. To this end, let us denote

ρ i (t ) = ui (t ) − vi (t ), t ≥ −τ max . Note that, for t ∈ [−τ max ,0] we have
ui (t ) ≤ φi − ξi

γ

≤ κ*  φ
*
m



≤ κ*  φ


∞

−

∞

−

γ 
ξ
δ *  i

γ  −σ t
ξ e = vi (t ) .
δ *  i

Thus, ρi (t) ≤ 0, for all t ∈ [ −τ max , 0], i ∈ n. Assume that there exist an index all i ∈ n
and all t1 > 0 such that ρ i (t1 ) = 0, ρ i (t ) > 0, t ∈ (t1 , t1 + δ ) for some δ > 0 and ρ j (t ) ≤ 0,
t ∈ [ −τ max , t1 ]. Then D + ρi (t1 ) > 0. However, for t ∈ [0, t1 ), it follows from (8) and (9) that
n

D + ρi (t1 ) ≤ a ii ρi (t ) +

∑

j =1, j ≠ i

n

a ij ρ j (t ) + ∑ bij ρ j (t − τ (t ))
j =1

≤ a ii ρ i (t ),
and therefore, D + ρi (t1 ) ≤ 0 which yields a contradiction. This shows that ρ i (t ) ≤ 0, for all
t1 > 0, i ∈ n. Consequently,


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xi (t ) ≤ ξi

21

γ

+κ*  φ
*
m


≤

γ
ξ
m*

≤

γ

+κ*  φ

*
m


∞

∞


+κ*  φ

∞

−

γ  −σ t
ξe
δ *  i

−

∞

−

γ 
ξ
δ * 

∞

e −σ t

γ  −σ t
e , t ≥ 0, i ∈ n.
δ * 

Finally, we obtain
xi (t )

Step 4. Let µ >

∞

≤

γ

+κ*  φ
*
m


∞

−

γ  −σ t
e , t ≥ 0.
δ * 

γ
and x (t , φ ) be any solution of system (1). If
m*

shown in Step 2, x(t , φ )

∞

(10)

φ

∞

≤

≤ µ holds for all t ≥ T ( µ , φ ) = 0. Assume that φ

γ
then, as
m*
∞

>

γ
then
m*

from (10) we have
x (t , φ )

≤

Therefore, if φ

∞

∞

≤

γ  φ ∞ γ  −σ t
+
− e
m*  ξ min δ * 

φ
γ
(1 − e−σ t ) + ∞ e −σ t .
*
ξ min
m

≤ µξ min , note that µ >
x (t , φ )

If φ

∞

∞

γ
then
m*

≤ µ (1 − e −σ t ) + µ e −σ t = µ , ∀ t ≥ 0.

> µξ min then
 φ∞ γ
− *

1
m
ξ
T (t , φ ) ≜ ln  min
γ

σ
 µ − m*


and x (t , φ )

∞




>0




≤ µ for all t ≥ T ( µ , φ ). This shows that system (1) is µ -practically stable.

The proof is completed.

⧠


Ha Noi Metroplolitan University

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Remark 3.2. It can be seen in the proof of Theorem 3.1 that (using (3) and (6)), for a
fixed vector ξ ∈ int ( ℝ n+ ) , satisfying
(A + B )ξ ≪ 0,

(11)

the exponential convergence rate σ = min i∈n σi , where σi is the unique positive solution of
the scalar equation

(

)

n

a ii + σ ξi +

∑

j =1, j ≠ i

n

a ijξ j + ∑ bij eστ max ξ j = 0.

(12)

j =1

Theorem 3.1 actually provides an explicit delay-independent criterion for the practical
exponential convergence of linear time-varying system (1). Moreover, the impact of timedelay on the decay rate is also explicit provided by computing the associated scalar σ in
(12) for any ξ ∈ int (ℝ n+ ) satisfying (11).

Remark 3.3. As an application to the linear time-varying systems without
disturbances (i.e. d(t) = 0 for all t), the proposed conditions in Theorem 3.1 guarantee the
Lyapunov exponential stability of the system as presented in the following corollary.
Corollary 3.1. Let assumption (A1) hold and assume that there exists a vector

ξ ∈ int (ℝ n+ ) such that ( A + B ) ξ ≪ 0. Then, system (1) without disturbance is exponentially
stable in the sense of Lyapunov. Moreover, any solution x(t, φ ) of satisfies
x (t )

∞

≤

ξ ∞
φ ∞ e −σ t , t ≥ 0,
ξ min

where σ = min i∈n σ i and σ i be the unique positive solution of the equation


  n 1

1
a
+
a
ξ
bijξ j  eστ max + σ = 0.
ii
ij

∑
j +∑
j ≠i ξi

  j =1 ξi

Remark 3.4. Corollary 3.1 gives a testable delay-independent condition for the global
exponential stability of linear time-varying systems with time-varying delay. This result
extends existing results, for example, in [31, 32], to time-varying systems.
Remark 3.5. As discussed in [11], the use of LKF method is only possible to achieve

condition for improving the exponential decay rate σ , the µ -neighborhood and the
transient time T ( µ , φ ) by using a special classes of functional v( xt ) leading exclusively to a
bounding of the form


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2 xT (t ) Pd (t ) ≤ 2
where α1 > 0, v( xt ) ≥ α1 x(t )

2

23

γ P

α1

v( xt ),

and P is symmetric positive definite matrix satisfying some

LMIs. However, the approach of [11] is not applicable to time-varying systems as
described by (1). In contrast, our approach directly uses the upper bound of disturbances to
obtain a bound on the system states.
As a brief discussion, we would like to mention here that, it is possible to derive the
exponential decay rate σ , the µ -neighborhood and the transient time T by imposing in
one condition is that the matrix Mσ = A + σ I + eστ max B is Hurwitz for some σ > 0. Then

µ and T can be determined as follows

Step 1. Find a vector ξ ∈ int (ℝ n+ ) , such that Mσ ξ ≪ 0 .
Step 2. Compute m* = ( N σ ξ )min and δ * = m* / ξ min , where N σ = σ I + (eστ max − 1)B .
Step 3. The µ -neighborhood and transient time T ( µ , φ ) are determined by µ >

 φ
γ 

 ∞ − * ξ
 1   ξ min m 
 ln
γ
T (µ , φ ) = σ 
 µ− * ξ ∞

m



0

∞









if φ

∞

>

γξ
m*

∞

and

ξ min
µ
ξ ∞

elsewhere.

4. SIMULATIONS
In this section, we give some numerical examples to illustrate the effectiveness of the
proposed conditions.
Example 4.1. Consider system (1), where
 cos 3t
cos 2t
 −3(1+ | sin t |


A(t ) = 
 , B(t ) =  sin t

−t
−t
2
−
e
−
5(1
+
e
cos
t
)


1+ | cos t |

It is easy to see that assumption (A1) is satisfied. In this case, we have:
 −3

A= 
1
and therefore

1
1 2 
,B =


−5 
1 0.5 

2sin t 
t sin t  .
1 + t 2 


Ha Noi Metroplolitan University

24

3 
 −2
M=A+B = 
.
 2 −9 / 2 
It is easy to verify that

ξ = (1 0.5)T ∈ int ( ℝ 2+ )
satisfies M ξ ≪ 0 . By Theorem 3.1, system (1) is practically stable. Taking (2) and (6)
*
*
*
into account we obtain m = 0.5, δ = 1, κ = 1 and σ = 0.1579 . Let γ = 0.1, that is, the

disturbance "
satisfies the threshold d (t ) ∞ ≤ 0.1. Then, any solution of system (1)
satisfies the following exponential practical estimation
x (t , φ )

Some

state

∞

≤ 0.2 + 2 ( φ

trajectories

of

(1)

+

∞

− 0.1) e −0,1579t , t ≥ 0.

with

d1 (t ) = 0.1sin 3 2t , d 2 (t ) = 0.1cos 4t and

τ (t ) =| sin( t ) | and τ (t ) =| sin( t ) | are presented in Fig. 1, which support the obtained
theoretical results.

Figure 1: State trajectories of system (1)

Example 4.2. Consider the following LTI system
ɺ = A 0 x(t) + A1x(t − τ) + d(t),

x(t)

(13)

8 
 −0.51 6 
0
, A1 = 
, τ ≥ 0 is a constant delay and d(t) is the
where A0 = 

−3 
−1
 0
 0 10 
disturbance input. By the results of [11], system (13) is practically stable if the following
LMI is feasible for symmetric positive definite matrices P, Q and a positive scalar σ [11]


Scientific Journal − No27/2018

 A0T P + PA0 + Q + 2σ P
PA1 

 < 0.
T
A1 P
−e −2στ Q 


25

(14)

Let us take r σ = 0.5 then, by using Matlab LMI toolbox, we found that (14) is
feasible with τ < 11.495.

 −0.01

68e0.5τ
However, for this example, we have Mσ = 
. Obviously, Mσ
−3 0.5τ 
−0.5 + 10 e 
 0
is Hurwitz if and only if 0.5 − 10 −3 e 0.5τ > 0, which allows a larger range of delay as

τ < 12.4292.

5. CONCLUSION
This paper has addressed the problem of practical stability of linear time-varying
systems with a time-varying delay and bounded disturbances. New explicit conditions have
been derived for determining a µ -neighborhood and a finite transient time T guaranteeing
that all state trajectories of the system converge exponentially to the µ -neighborhood after
the transient time T.

REFEENCES
1.

V. Kolmanovskii and A. Myshkis (1992), “Applied Theory of Functional Differential

Equations’’, Dordrecht: Kluwer AP.

2.

S.I. Niculescu (2001), “Delay Effects on Stability: A Robust Control Approach”, Berlin:
Springer-Verlag.

3.

K. Gu and S.I. Niculescu (2003), “Survey on recent results in the stability and control of timedelay systems”, J. Dyn. Syst., Meas., and Control, vol. 125, N0.2, pp.158-165.

4.

S. Mondié and V. Kharitonov (2005), “Exponential estimates for time delay systems: a LMI
approach”, IEEE Trans. Autom. Control, vol. 50, N0. 2, pp.268-273.

5.

J.J. Loiseau,W. Michiels, S.I. Niculescu and R. Sipahi (2009), “Topics in Time Delay
Systems: Analysis, Algorithms and Control”, Berlin: Springer-Verlag.

6.

L.V. Hien and V.N. Phat (2009), “Exponential stability and stabilization of a class of uncertain
linear time-delay systems”, J. Frankl. Inst., vol. 346, N0.6, pp.611-625.

7.

H. Trinh, P. Teh and T. Fernando (2010), “Time-delay systems: Design of delay-free and loworder observers”, IEEE Trans. Autom. Control, vol. 55, N0.10, pp.2434-2438.

8.

R. Sipahi, S.I. Niculescu, C.T. Abdallah, W. Michiels and K. Gu (2011), “Stability and
stabilization of systems with time delay”, IEEE Control Syst., vol. 31, N0.1, pp.38-65.

9.

P.H.A. Ngoc (2013), “Stability of positive differential systems with delay”, IEEE Trans.
Autom. Control, vol. 58, N0.1, pp.203-209.


26

Ha Noi Metroplolitan University

10. Y. Li, K. Gu, J. Zhou and S. Xu (2014), “Estimating stable delay intervals with a discretized
Lyapunov-Krasovskii functional formulation”, Automatica, vol. 50, N0.6, pp.1691-1697.
11. R. Villafuerte, S. Mondié and A. Poznyak (2011), “Practical stability of time-delay systems:
LMI’s approach”, Eur. J. Control, vol. 17, no. 2, pp. 127-138.
12. M. Corless and G. Leitmann (1993), “Bounded controllers for robust exponential
convergence”, J. Optim. Theory Appl., vol. 76, N0.1, pp.1-12.
13. T. Hu and Z. Lin (2001), “Control Systems with Actuator Saturation: Analysis and Design”,
Boston: Birkh-@ user.
14. A. Poznyak, V. Azhmyakov and M. Mera (2011), “Practical outputfeedbackstabilisation for a
class of continuous-timedynamic systems under sample-data outputs”, Int. J. Control, vol. 84,
N0.8, pp.1408-1416.
15. R. Kuiava, R.A. Ramos, H.R. Pota and L.F.C. Alberto (2013), “Practical stability of switched
systems without a common equilibriaand governed by a time-dependent switching signal”,
Eur. J. Control, vol. 19, N0.3, pp.206-213.
16. I. Karafyllis, M. Krstic, T. Ahmed-Ali and F. Lamnabhi-Lagarrigue (2014), “Global

stabilisation of nonlinear delay systems with a compact absorbing set”, Int. J. Control, vol. 87,
N0.5, pp.1010-1027.
17. H. K. Khalil (2002), “Nonlinear Systems”, New York: Prentice Hall.
18. J. LaSalle and S. Lefschetz (1961), “Stability by Liapunov's Direct Method with
Applications”, New York: Academic Press.
19. V. Lakshmikantham, S. Leela and A.A. Martynyuk (1990), “Practical Stability of Nonlinear
Systems”, Singapore: World Scientific.
20. L. Moreau and D. Aeyels (2000), “Practical stability and stabilization”, IEEE Trans. Autom.
Control, vol. 45, N0.8, pp.1554-1558.
21. V. Lakshmikantham and Y. Zhang (2001), “Strict practical stability of delay differential
equation”, Appl. Math. Comput., vol. 122, N0.3, pp.341-351.
22. X. Xu and G. Zhai (2005), “Practical stability and stabilization of hybrid and switched
systems”, IEEE Trans. Autom. Control, vol. 50, N0.11, pp.1897-1903
23. S. Li, X. Song and A. Li (2008), “Strict practical stability of nonlinear impulsive systems by
employing two Lyapunov-like functions”, Nonlinear Anal. Real World Appl., vol. 9, N0.5,
pp.2262-2269.
24. X. Chen and Y. Zou (2010), “Generalized practical stability analysis of Filippov-type
systems”, J. Math. Anal. Appl., vol. 367, N0.1, pp.304-316.
25. L. Wang, L. Cai, X. Liu and X. Shen (2010), “Bounds estimation and practical stability of
AIMD/RED systems with time delays”, Computer Netw., vol. 54, N0.7, pp.1069-1082.
26. R. Villafuerte, B. Saldivar and S. Mondié (2013), “Practical stability and stabilization of a
class of nonlinear neutral typetime delay systems with multiple delays: BMI's approaches”,
Int. J. Control Autom., vol. 11, N0.5, pp.859-867.


Scientific Journal − No27/2018

27

27. V.N. Phat and L.V. Hien (2009), “An application of Razumikhin theorem to exponential

stability forlinear non-autonomous systems with time-varying delay”, Appl. Math. Lett., vol.
22, N0.9, pp.1412-1417.
28. L.V. Hien (2014), “An explicit criterion for finite-time stabilityof linear nonautonomous
systems with delays”, Appl. Math. Lett., vol. 30, pp.12-18.
29. L.V. Hien and H.M. Trinh (2014), “A new approach to state bounding for linear time-varying
systemswith delay and bounded disturbances”, Automatica, vol. 50, N0.6, pp.1735-1738.
30. A. Berman and R.J. Plemmons (1994), “Nonnegative Matrices in the Mathematical Sciences”,
Philadelphia: SIAM.
31. X. Liu, W. Yu and L. Wang (2010), “Stability analysis for continuous-time positive systems
with time-varying delays”, IEEE Trans. Autom. Control, vol. 55, N0.4, pp.1024-1028.
32. H. R. Feyzmahdavian, T. Charalambous and M. Johansson (2014), “Exponential stability of
homogeneous positive systems of degree one with time-varying delays”, IEEE Trans. Autom.
Control, vol. 59, N0.6, pp.1594-1599.

TÍNH ỔN ĐỊNH THỰC HÀNH
CỦA CÁC HỆ TRỄ TUYẾN TÍNH KHƠNG DỪNG
Tóm tắ
tắt: Bài báo này nghiên cứu tính ổn định thực hành đối với lớp hệ tuyến tính khơng
dừng với trễ biến thiên và nhiễu bị chặn. Dựa trên một số kĩ thuật so sánh trong lý thuyết
hệ dương, các điều kiện hiển, độc lập với độ trễ, được thiết lập cho việc xác định một lân
cận compact của điểm cân bằng lí tưởng (điểm gốc) bao chung cuộc mọi quỹ đạo trạng
thái của hệ.
Từ khóa:
khóa Tính ổn định thực hành, trễ biến thiên, ma trận Metzler.