Practical stability of linear time-varying delay systems

This paper is concerned with the problem of practical stability of linear time-varying 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.

PRACTICAL STABILITY OF LINEAR TIME

PRACTICAL STABILITY OF LINEAR TIME VARYING VARYING VARYING

DELAY SYSTEMS DELAY SYSTEMS

Le Van Hien

Hanoi National University of Education

Abstract:

Abstract: This paper is concerned with the problem of practical stability of linear

time-varying 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:

Keywords: Practical stability, time-varying delay, Metzler matrix

Email: hienlv@hnue.edu.vn

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

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

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 ℕdenote the set of real numbers and natural numbers, respectively For a given n ∈ ℕ, n≜{1,2,…, n}.ℝnis the n-dimensional vector space endowed with the norm ‖x‖ = max ∈ |xi | for x = (xi) ∈ ℝn The non-negative orthant of ℝn will be denoted by n

+

ℝ By int(X), we denote the interior of the subset n

X ⊂ ℝ Let ℝm n× be the set

of all m × n real matrices For a matrix m n

A∈ ℝ × , ( ) 1m

i

r A ∈ ℝ× , denotes the ith row of A Inequalities between vectors will be understood componentwise Specifically, for u = (u i)

and = ( i) inℝn , u ≥ means ≥ for all ∈ and if > for all ∈ then we write u ≫ instead of > In particular, n { n: 0}

int ℝ+ = x∈ℝ x≫ Denote min = min∈ i then > 0 for any vector

i

v= v ∈int ℝ+ . We also specifically use the notation = max{ , 0} for real number

, that means = if and only if > 0, otherwise = 0

Consider the following linear time-varying system with delay

max

( ) ( ) ( ) ( ) ( ( )) ( ), 0,

x t A t x t B t x t t d t t

x t t t

τ

ɺ

(1) where = ! ∈ℝnis the system state vector and " = " ! ∈ℝnis unknown input disturbance vector, ( ) ( ( )) n n

ij

A t = a t ∈ ℝ × and ( ) ( ( )) n n

ij

B t = b t ∈ ℝ× are time-varying system matrices whose elements are assumed to be continuous on ℝ+, ( )τ t is a time-varying delay and (.) ([ max, 0], n)

C

φ ∈ −τ ℝ is the vector-valued initial function specifying the initial state of the system, ( ) ( ( )) n

i

φ = φ ∈ ℝ Let us denote

| |i sup | ( ) |i

t

t

τ

= and φ ∞=maxi n∈ | |φi

Remark 2.1 In this paper, the time delay # is assumed to be continuous in time, not necessarily differentiable, and satisfies 0 ≤ # ≤ # %&, for all ≥ 0, where the upper bound # %& is a known constant We do not impose any restriction on the rate of change of # (such as slowly time-varying condition #( ≤ #)<1 for all ≥ 0 This 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 (.) ([ max, 0], n)

C

φ ∈ −τ ℝ

these exists a transient time T = T( , ) 0µ φ ≥ 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 n n

M∈ ℝ× is said to be Metzler matrix if all off-diagonal elements of

M are nonnegative, i.e., if ( ) n n

ij

M = a ∈ ℝ × is a Metzler matrix then -.≥ 0 for all ≠ 0 For a matrix M∈ℝn n× , the spectrum of M is defined asσ( ) {M = λ∈ℂ: det(λI n−M) 0}=

and the spectral abscissa of M is given by ( ) max{Re :µ M = λ λ σ∈ ( )}M We now summarize some properties of Metzler matrices in the following proposition

Proposition 2.1 Let n n

M∈ ℝ × be a Metzler matrix The following statements are equivalent

(i) µ( ) 0.M <

(ii) M is invertible and 123≤ 0

(iii) There exists a vector ( n)

int

ξ∈ ℝ+ such that Mξ ≪0 (iv) For any b∈int(ℝn+), there exists x ∈ ℝ such that Mx + b = 0 n+

(v) There exists a vector ( n)

int

η∈ ℝ+ such that T 0

M η≪ (vi) For any ∈ ℝn+\ {0},

x the row vector 41 has at least one negative entry

From Proposition (1), we obtain the following result

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

int

ξ∈ ℝ+ , ξ ∞ =1, such that Mξ ≪0

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

lim ( ) ( ) ( ) sup ,

h

v t h v t

h

+ +

→

+ −

3 MAIN RESULTS

The following assumptions will be used in the derivation of our results

(A1) The system matrices ( ) ( ( )) n n

ij

A t = a t ∈ ℝ× and ( ) ( ( )) n n

ij

B t = b t ∈ ℝ × satisfy the following conditions

- ≤ -5 , 6-. 6 ≤ -5 , ≠ 0, 67. 6 ≤ 75., ∀ ≥ 0, , 0 ∈

(A2) The disturbance vector " = " is bounded, that means, there exists a positive constant γ such that

|" | ≤ γ , ∀ ≥ 0, ∈

Remark 3.1 For any initial function (.) ([ max, 0], n)

C

φ ∈ −τ ℝ , there exists a unique solution x t ( , ) φ of (1) defining on [ − τmax, ] ∞ [1] On the other hand, according to (A2),

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 denoteA =( )a ij , B=( )b ij and M A B = + Clearly, Mis a Metzler matrix Therefore, ifMsatisfies one of the equivalent conditions in Proposition 2.1 then, by Proposition 2.2, there exists a vectorξ∈int(ℝn+), 1

ξ ,such that Mξ ≪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

*-practically stable for any * > γ∗ The transient time ( , ), T µ φ (.) ([ max, 0], n)

C

given by

* min

min

*

1

( , )

∞

∞

−





m T

m

φ γ ξ

φ µξ γ

µ φ σ µ

Here ξ ∈int( )R+n is a vector satisfying ξ ∞=1 and Mξ ≪0, * ( )

min

m = −Mξ ,

*

*

min

m

δ

ξ

= and σ =mini n∈ σi, where σi is the unique positive solution of the scalar equation

1

n

j

σξ ξ

=

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

+

−

t

m

σ

δ

where *

min

κ = ξ

Proof We divide the proof into several steps

Step 1 By Proposition 2.2, there exists a vector ( n)

int

ξ∈ ℝ+ , ξ ∞ =1, such that 0

ξ ≪

M For convenience, we denoteD=diag{ }a ii and AD =A−D Then, we have

Note that

m = −Mξ = ∈ −aξ −r A +B ξ >

and from (2) we have

*

ii i i D

Step 2 We will prove that, if γ*,

m

φ ∞ ≤ then x t( , ) γ*,

m

φ ∞ ≤ for all ≥ 0 In the

following, we will use to denote the solution ( , )x tφ if it does not make any confusion

m

φ ∞≤ then we have x t i( ) i i γ*,

m

φ ξ

≤ ≤ for all t∈ −[ τmax;0],i n∈ For any given < > 1, assume that there exists an index i∈n and t>0such that *

γ ( )

m

ξ

= and

*

γ

( ) , [0, ],

m

ξ

≤ ∈ j∈ Then,n D x t+ i( ) ≥0. On the other hand, it follows from (1) that

1,

1

( ) sgn( ( )) ( )

( ) | ( ) | | ( ) || ( ) |

| ( ) || ( ( )) | | ( ) |

n

j j i n

j

+

= ≠

=

=

∑

1,

( ) | ( ) | n ( ) | ( ) |

j j i

= ≠

1

( ) | ( ( )) | γ,

n

j

b t x t τ t

=

+∑ − + t∈[0, ].t (4) Thus,

*

γ ( ) ( ( ) γ (1 ) 0

ij

q

m q

+

≤ − <

ξ γ

(5)

which yields a contradiction This shows that γ*

( ) ,

m

ξ

≤ for all t ≥0,i∈n Let < ↓ 1

( ) ,

x t

m

ξ

≤ for all i∈n and t≥0, and hence,

( ) , t 0

i

x t

Step 3 Now, we assume γ*

φ δ

∞ > Then, it is easy to verify that

*

,

m

δ

∞

≤ ≤  −  ∈

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

ij 1,

j j i

σ σξ ξ (6)

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 σi In addition, ( ) 0H i σ < for all σ∈(0, ].σi Let σ = mini n∈ σithen H i( ) 0σ ≤ for all i∈n

Let us consider the functions all ( ), v ti i ∈ n , defined as follows

*

max

*

δ

−

∞

For anyt ≥ 0 and j∈n, it is clear from (7) that

*

τ κ φ ξ

δ

∞

* max

*

t

i e σeστ

γ

κ φ ξ

δ

−

∞

≤  −  ≤eστmaxv t i( )

( ) ( ) n ( ) n ( ( ))

+ ∑ +∑ −

max

t

( max )

1

t

j

=

σ

( max )

* 1

1

n t

ij j j

=

≤ − + − 

, 0, i ,

t

βσξ −

≤ − ≥ ∈

*

γ

β κ φ

δ

∞

=  − 

The above estimation leads to

( ) ii ( ) n ii ( ) n ij ( ( )), 0

v t a v t a v t b v t t t

Next, by using the following transformation

γ ( ) ( ) , , ,

m

= − ≥ − ∈

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

( ) ii ( ) n ij ( ) n ij( ) ( ( ))

(a ii i r i D )

m

γ

+ + A +B + (9)

( ) 1, ( ) 1 ( ) ( ( )),

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

max

i t u t i v t t i

ρ = − ≥ −τ Note that, for t∈ −[ τmax,0] we have

*

( )

u t

m

φ ξ κ φ ξ

δ

∞

*

*

i e σ v t i

κ φ ξ

δ

−

∞

Thus, ρi(t) 0,≤ for all t∈ −[ τmax, 0],i ∈ Assume that there exist an index all i n ∈ n

and all t1> such that 0 ρi( ) 0, ( ) 0,t1 = ρi t > t∈( ,t t1 1+δ) for some δ > 0 and ρj( ) 0,t ≤

t∈ −τ t ThenD+ρi( ) 0.t1 > However, for t∈[0, ),t1 it follows from (8) and (9) that

( ) ii ( ) n ij ( ) n ij ( ( ))

D+ρ t a ρ t a ρ t b ρ t τ t

≤a iiρi( ),t

and therefore, D+ρi( ) 0t1 ≤ which yields a contradiction This shows that ρi( ) 0,t ≤ for all

t > i∈n Consequently,

*

m

σ

ξ κ φ ξ

δ

−

∞

*

e m

σ

ξ κ φ ξ

δ

−

*

t

m

σ

κ φ

δ

−

∞

Finally, we obtain

*

i

m

σ

κ φ

δ

−

(10)

Step 4 Let γ*

m

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

m

φ

∞≤ then, as shown in Step 2, x t( , )φ ∞ ≤µ holds for allt≥T( , ) 0.µ φ = Assume that γ*

m

φ ∞ > then from (10) we have

m

σ

φ φ

ξ δ

−

∞

∞

γ

m

ξ

Therefore, if φ ∞≤µξmin, note that γ*

m

µ> then

x t φ µ e−σ µe−σ µ t

If φ ∞ >µξminthen

* min

*

γ 1

γ

m

T t

m

φ ξ φ

σ µ

∞

−

−

≜

and x t( , )φ ∞ ≤µ for all t≥T( , ).µ φ This shows that system (1) is µ-practically stable The proof is completed ⧠