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Automatica

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Technical communique

A new approach to state bounding for linear time-varying systemswith delay and bounded disturbances✩

Le Van Hien a, Hieu Minh Trinh b,1

a Department of Mathematics, Hanoi National University of Education, 136 Xuan Thuy Road, Hanoi, Viet Namb School of Engineering, Deakin University, Geelong, VIC 3217, Australia

a r t i c l e i n f o

Article history:Received 3 September 2013Received in revised form22 January 2014Accepted 17 April 2014Available online xxxx

Keywords:State boundingReachable setTime-varying systemsBounded disturbancesMetzler matrix

a b s t r a c t

In this note, the problem of state bounding for linear time-varying systems with delay and bounded dis-turbances input is considered for the first time. By using a novel approach which does not involve theLyapunov–Krasovskii functional method, new explicit delay-independent conditions are derived for theexistence of a ball such that all the state trajectories of the system converge exponentially within it. Anumerical example is given to illustrate the effectiveness of the obtained result.

© 2014 Elsevier Ltd. All rights reserved.

1. Introduction

In the last decade, we have witnessed an increasing interestto the problem of state bounding, which includes reachable setbounding and state convergence, for linear time-delay systems(see, Fridman & Shaked, 2003; Kim, 2008; Kwon, Lee, & Park,2011; Lam, Zhang, Chen, & Xu, 2013; Nam & Pathirana, 2011;Nam, Pathirana, & Trinh, 2013; Shen & Zhong, 2011; That, Nam,& Ha, 2013; Zuo et al., 2013; Zuo, Ho, & Wang, 2010). Themethods which are most commonly used in the existing resultsare based on the Lyapunov–Krasovskii functionals or the Lya-punov–Razumikhin technique. Particularly, Fridman and Shaked(2003) derived delay-dependent conditions for determining an el-lipsoid that contains the reachable set for a linear time-delay sys-tem via the Lyapunov–Razumikhin approach. Zuo et al. (2013,2010) constructed a point-wise maximum functional of a family

✩ This paper was supported in part by the Australian Research Council (ARC)(DP130101532), the NAFOSTED of Vietnam (101.01-2011.51) and the Ministry ofEducation and Training of Vietnam (B2013.17.42). The material in this paper wasnot presented at any conference. This paper was recommended for publication inrevised form by Associate Editor Keqin Gu under the direction of Editor André L.Tits.

E-mail addresses: hienlv@hnue.edu.vn (L.V. Hien), hieu.trinh@deakin.edu.au(H.M. Trinh).1 Tel.: +61 3 52272030; fax: +61 3 52272167.

of Lyapunov–Krasovskii functionals to derive reachable set condi-tions for polytopic systems. Recently, Lam et al. (2013) extendedthe ideas of reachable set estimation of continuous-time systemsto discrete-time systems with multiple constant delays. That et al.(2013) used an improved Lyapunov–Krasovskii functional combin-ing with the delay-decomposition technique to derive a reachableset estimation defined as an intersection of a family of ellipsoids fora class of discrete-time systems with interval time-varying delay.

However, all of the aforementioned works on the state bound-ing have been mainly focused on linear time-delay systems withconstantmatrices or a combination of constantmatrices (polytopicuncertainties). So far, there has not been any paper that deals withthe problemof state bounding for time-varying systemswith delayand bounded disturbances. It should be noted that, when treatingtime-varying systems with delay, the developed methodologiessuch as Lyapunov–Krasovskii functional cannot be used becausethey either lead to matrix Riccati differential equations (RDEs) orindefinite LMIs as there is no efficient computational tool availableto solve them. Since an ellipsoidal estimation of the reachable setcan only be defined by a certain class of Lyapunov–Krasovskii func-tionals, it is no longer suitable for time-varying systems. Clearly,there is a need for an alternative approachwhendealingwith time-varying systems.

In this note, for the first time, we consider the problem of statebounding for linear time-varying systems with time-varying delayand bounded disturbances. By a novel approach, without using the

http://dx.doi.org/10.1016/j.automatica.2014.04.0250005-1098/© 2014 Elsevier Ltd. All rights reserved.

2 L.V. Hien, H.M. Trinh / Automatica ( ) –

Lyapunov–Krasovskii functional method, we derive explicit delay-independent conditions in terms of theMetzlermatrix for the exis-tence of a ball which bounds all the state trajectories of the systemfor some domain of initial conditions. For the other initial condi-tions, the derived conditions ensure that all the state trajectoriesof the system converge exponentially within the ball.Notations. We denote n := {1, 2, . . . , n}. Rn denotes the n-dimen-sional space with the norm ∥x∥∞ = maxi∈n |xi|. Rm×n denotes theset of real m × n-matrices. For u = (ui), v = (vi) in Rn, u ≥ v iffui ≥ vi, ∀i ∈ n; u ≫ v iff ui > vi, ∀i ∈ n. Rn

+= {x ∈ Rn: x ≥ 0}. For

a given r > 0, BFr([0, ∞), Rm) = {w: [0, ∞) → Rm| ∥w(t)∥∞ ≤

r, ∀t ≥ 0} is the set of Rm-valued functions bounded by r . For avector ξ ∈ Rn

+, we denote (ξ)min = mini∈n ξi.

2. Problem statement and preliminaries

Consider the following linear time-varying system with delay

x(t) = A(t)x(t) + D(t)x(t − τ(t)) + B(t)w(t), t ≥ 0,x(t) = φ(t), t ∈ [−τM , 0],

(1)

where x(t) ∈ Rn is the state vector, w(t) ∈ Rm is the externaldisturbance vector, A(t) = (aij(t)) ∈ Rn×n,D(t) = (dij(t)) ∈ Rn×n

and B(t) = (bij(t)) ∈ Rn×m are system matrices. The disturbancew(t) is unknown but is assumed to be bounded by a given constantw ≥ 0, i.e., w ∈ BFw([0, ∞), Rm). The time-varying delay τ(t)satisfies 0 ≤ τ(t) ≤ τM , t ≥ 0. φ(t) = (φi(t)) ∈ C([−τM , 0], Rn)is the initial condition. Let us denote |φi| = sup−τM≤t≤0 |φi(t)| and∥φ∥∞ = maxi∈n |φi|.

For ϵ > 0, let the ball B(ϵ) be defined by B(ϵ) = {x ∈

Rn: ∥x∥∞ ≤ ϵ}. By adopting the concept of ball convergence inOucheriah (2006), we have the following definition.

Definition 1. For a given ϵ > 0. System (1) is said to be globallyexponentially convergent within the ball B(ϵ) if there exist aconstant γ > 0 and a non-decreasing functional κ(.) such thatthe following inequality holds

∥x (t, φ,w) ∥∞ ≤ ϵ + κ (∥φ∥∞) e−γ t , t ≥ 0,

for all φ ∈ C([−τM , 0], Rn) and w ∈ BFw([0, ∞), Rm).

The main objective of this note is to derive conditions for theexistence of the balls B(ϵ1), B(ϵ2) such that: (i) for all initialconditions in B(ϵ1), the corresponding trajectories of the systemare bounded by the B(ϵ2) for all time; and (ii) for the otherinitial conditions, which are outside B(ϵ1), the correspondingtrajectories of the system converge exponentially within B(ϵ2).

At first, we recall here some properties of Metzler matrix (see,Berman & Plemmons, 1994 for more details). For any matrix M ∈

Rn×n, the spectrum of M is denoted by σ(M) = {λ ∈ C: det(λIn −

M) = 0} and the spectral abscissa of M is denoted by µ(M) =

max{Reλ: λ ∈ σ(M)}. A matrixM ∈ Rn×n is called aMetzler matrixif all off-diagonal elements of M are non-negative. The followingproposition is used in stating our main results.

Proposition 2. Let M ∈ Rn×n be aMetzler matrix. Then the followingstatements are equivalent(i) µ(M) < 0.(ii) M is invertible and M−1

≤ 0.(iii) There exists ξ ∈ Rn, ξ ≫ 0 such that Mξ ≪ 0.(iv) For any b ∈ Rn, b ≫ 0, there exists x ∈ Rn

+such that Mx+b = 0.

(v) There exists η ∈ Rn, η ≫ 0 such that MTη ≪ 0.(vi) For any x ∈ Rn

+\{0}, the row vector xTM has at least one negative

entry.

3. Main results

Let A(t) = (aij(t)),D(t) = (dij(t)) and B(t) = (bij(t)) are givenmatrices with continuous elements. Assume that

(A1) aii(t) ≤ aii, i ∈ n, |aij(t)| ≤ aij, i = j, i, j ∈ n, t ≥ 0.(A2) |dij(t)| ≤ dij, |bij(t)| ≤ bij, ∀t ≥ 0, i, j ∈ n.

We denote A = (aij) and D = (dij). To facilitate in presentingour main objective as discussed before, we use the notation α+

=

max{α, 0} for a real number α. That means α+= α iff α > 0,

otherwise α+= 0.

We are now in the position to state our main result as follows.

Theorem 3. Let assumptions (A1), (A2) hold and the matrix M =

A + D satisfies one of the equivalent conditions (i)–(vi) ofProposition 2. Then there exist positive constants β, γ , η∗ and δ∗ suchthat the following estimation holds for all φ ∈ C([−τM , 0], Rn) andw ∈ BFw([0, ∞), Rm)

∥x (t, φ,w) ∥∞ ≤b∞

η∗w + β

∥φ∥∞ −

b∞

δ∗w

+

e−γ t , t ≥ 0,

where b∞ = maxi∈nm

j=1 bij.

Remark 4. To deal with time-varying systems, in the proof ofTheorem 3, we develop the comparison principle for estimating|xi(t)|, i ∈ n, in which some similar arguments to the Razumikhinapproach will be used.Proof. By (A1) and (A2), M is a Metzler matrix. Therefore, anyof two conditions (i)–(vi) of Proposition 2 are equivalent. Letcondition (iii) of Proposition 2 hold. Then there exists ξ ∈ Rn, ξ ≫

0 such that Mξ ≪ 0, and thus,n

j=1

aij + dij

ξj < 0, ∀i ∈ n. (2)

Note that, in (2), ξj can be replaced by ξj/∥ξ∥∞, and then, withoutloss of generality, we can assume that ∥ξ∥∞ = 1. We define thefollowing constants

η∗= min

i∈n

−

nj=1

aij + dij

ξj

, δ∗

=η∗

(ξ)min, (3)

then η∗ > 0 andn

j=1

aij + dij

ξj ≤ −η∗ for all i ∈ n. Let

ϵ1 =b∞

δ∗ w and ϵ2 =b∞

η∗ w. We first prove that, if ∥φ∥∞ ≤ ϵ1

then ∥x(t, φ,w)∥∞ ≤ ϵ2 for all t ≥ 0. In the following, we willuse x(t) to denote the solution x(t, φ,w) if it does not cause anyconfusion. Let ∥φ∥∞ ≤ ϵ1 then we have |xi(t)| ≤ |φi| ≤ ξiϵ2 forall t ∈ [−τM , 0] and i ∈ n. Assume that there exists an index i ∈ nand t > 0 such that |xi(t)| = ξiϵ2 and |xj(t)| ≤ ξjϵ2, ∀t ≤ t, j ∈ n.Then we have

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

≤ aii(t)|xi(t)| +

nj=1,j=i

|aij(t)| |xj(t)|

+

nj=1

|dij(t)| |xj(t − τ(t))| +

mj=1

|bij(t)| |wj(t)|

≤ aii|xi(t)| +

nj=1,j=i

aij|xj(t)|

+

nj=1

dij|xj(t − τ(t))| +

mj=1

bijw, t ∈ [0, t], (4)

where D+ denotes the upper-right Dini derivative. Thus,

D+|xi(t)| ≤ ϵ2

nj=1

(aij + dij)ξj

+ b∞w

≤ −ϵ2η∗+ b∞w ≤ 0, (5)

L.V. Hien, H.M. Trinh / Automatica ( ) – 3

due to (3). It follows from (5) that |xi(t)| cannot exceed ξiϵ2 for allt ≥ 0. Therefore, |xi(t)| ≤ ξiϵ2 for all i ∈ n and t ≥ 0, and hence,∥x(t)∥∞ ≤ ϵ2∥ξ∥∞ = ϵ2, ∀t ≥ 0.

Next, assume that ∥φ∥∞ −b∞

δ∗ w > 0. For each i ∈ n, considerthe following equation

Hi(γ ) = γ ξi +

nj=1

ξjdij (eγ τM − 1) − η∗= 0, γ ∈ [0, ∞). (6)

SinceHi(γ ) is continuous and strictly increasing on [0, ∞),Hi(0) <0,Hi(γ ) → ∞, γ → ∞, Eq. (6) has a unique positive solution γi.Let γ = mini∈n γi then Hi(γ ) ≤ 0 for all i ∈ n. We define a positiveconstant β satisfying

|φi| − ξib∞

η∗w ≤ β

∥φ∥∞ −

b∞

δ∗w

ξi, ∀i ∈ n, (7)

and consider the functions vi(t), i ∈ n, as follows:

vi(t) = β

∥φ∥∞ −

b∞

δ∗w

ξie−γ t , t ≥ −τM .

It is noted that, for all t ≥ 0 and j ∈ n, we have

vj(t − τ(t)) = β

∥φ∥∞ −

b∞

δ∗w

ξje−γ (t−τ(t))

≤ β

∥φ∥∞ −

b∞

δ∗w

ξje−γ teγ τM ≤ eγ τM vj(t).

Therefore, by using (3) and (6)

aiivi(t) +

nj=1,j=i

aijvj(t) +

nj=1

dijvj(t − τ(t))

≤ κe−γ taiiξi +

nj=1,j=i

aijξj +n

j=1

ξjdijeγ τM

,

≤ κe−γ t n

j=1

aij + dij

ξj +

nj=1

ξjdij (eγ τM − 1)

≤ κe−γ t−η∗

+

nj=1

ξjdij (eγ τM − 1)

≤ −κγ ξie−γ t , t ≥ 0, ∀i ∈ n, (8)

where κ = β∥φ∥∞ −

b∞

δ∗ w. Thus, from (8) we obtain

vi(t) ≥ aiivi(t) +

nj=1,j=i

aijvj(t) +

nj=1

dijvj(t − τ(t)), t ≥ 0. (9)

Let us use the following transformations

ui(t) = |xi(t)| − ξib∞

η∗w, t ≥ −τM , i ∈ n. (10)

For i ∈ n, by combining (4) and (10), we have

D+ui(t) ≤

nj=1

aijuj(t) +

nj=1

dijuj(t − τ(t))

+b∞

η∗w

nj=1

aij + dij

ξj + b∞w

≤

nj=1

aijuj(t) +

nj=1

dijuj(t − τ(t)), t ≥ 0. (11)

We will prove that ui(t) ≤ vi(t), ∀t ≥ 0, i ∈ n. Let ρi(t) =

ui(t) − vi(t), t ≥ −τM . Note that, for t ∈ [−τM , 0], we have

ui(t) ≤ |φi| − ξib∞

η∗w ≤ β

∥φ∥∞ −

b∞

δ∗w

ξi

≤ β

∥φ∥∞ −

b∞

δ∗w

ξie−γ t

= vi(t).

Thus, ρi(t) ≤ 0, for all t ∈ [−τM , 0], i ∈ n. Assume that thereexists an index i ∈ n and t1 > 0 such that ρi(t1) = 0, ρj(t) ≤ 0,∀t ∈ [−τM , t1] and ρi(t) > 0, t ∈ (t1, t1 + δ) for some δ > 0. ThenD+ρi(t1) > 0. However, it follows from (9) and (11) that

D+ρi(t) ≤ aiiρi(t) +

nj=1,j=i

aijρj(t) +

nj=1

dijρj(t − τ(t))

≤ aiiρi(t), t ∈ [0, t1),

and therefore, D+ρi(t1) ≤ 0, which yields a contradiction. Thisshows that ρi(t) ≤ 0 for all t ≥ 0, i ∈ n. Consequently,

|xi(t)| ≤ ξib∞

η∗w + β

∥φ∥∞ −

b∞

δ∗w

ξie−γ t

≤b∞

η∗w∥ξ∥∞ + β

∥φ∥∞ −

b∞

δ∗w

∥ξ∥∞e−γ t

≤b∞

η∗w + β

∥φ∥∞ −

b∞

δ∗w

e−γ t , ∀t ≥ 0, i ∈ n.

Finally, we obtain

∥x(t)∥∞ ≤b∞

η∗w + β

∥φ∥∞ −

b∞

δ∗w

+

e−γ t , t ≥ 0.

The proof is completed. �

Remark 5. Note that, κ(∥φ∥∞) = β

∥φ∥∞ −

b∞

δ∗ w

+

is a

non-decreasing functional, and thus, (1) is globally exponentiallyconvergent within the ball B( b∞

η∗ w) for all disturbance w ∈

BFw([0, ∞), Rm). Moreover, the radius ϵ =b∞

η∗ w depends explic-itly on the upper bound of the disturbance which can be directlyapplied to the problem of global exponential stability of time-varying system without disturbances by letting w = 0.

Remark 6. It can be seen that the constant β = (ξ)−1min satisfies

(7) for all φ ∈ C([−τM , 0], Rn), ∥φ∥∞ > b∞

δ∗ w. Thus, the constantsβ, γ , η∗, δ∗ in the statement of Theorem 3 can be determined asfollows:

• Define a vector ξ ≫ 0 with ∥ξ∥∞ = 1 such that Mξ ≪ 0.• Compute η∗

= (−Mξ)min, δ∗= η∗(ξ)−1

min, β = (ξ)−1min.

• Maximizing γ > 0 satisfying Hi(γ ) ≤ 0 for all i ∈ n.

Corollary 7. Let assumptions (A1), (A2) hold and the matrix M =

A + D satisfies one of the equivalent conditions (i)–(vi) ofProposition 2. Then system (1)withw(t) = 0 is globally exponentiallystable. Moreover, every solution x(t, φ) of (1) satisfies the followinginequality

∥x(t, φ)∥∞ ≤ (ξ)−1min∥φ∥∞e−γ t , t ≥ 0,

where γ is defined by (6).

4. A numerical example

Consider the following linear time-varying system

x(t) = A(t)x(t) + D(t)x(t − τ(t)) + B(t)w(t), t ≥ 0, (12)

4 L.V. Hien, H.M. Trinh / Automatica ( ) –

Fig. 1. A state trajectory of (12) with w(t) = 0.5 sin 4t .

where x(t) ∈ R3, τ (t) = 6sin

2√t, and

A(t) =

−4 − |sin t|e−t cos 2t sin2 t

e−t cos t −6 + sin 2t 2 cos 3tt sin t1 + t

1√1 + |sin t|

−5 − |cos t|

,

D(t) =

sin 3t −e−2t 0e−2t sin t 0 cos 3t

0 cos2 t −e−t sin 2t

,

B(t) =

0.1esin t

0.2 cos 2t0.1 sin 4t

.

It can be seen that (A1) and (A2) are satisfied with A =−4 1 11 −5 21 1 −5

, D =

1 1 01 0 10 1 1

and M = A + D satisfies

condition (iii) of Proposition 2 with ξ T=

1 1 0.8

. From (3)

and (6) we have η∗= 0.2, δ∗

= 0.25 and γ = 0.0148. Letthe disturbance w satisfy |w(t)| ≤ w = 0.5 then all the statetrajectories of system (12) are inside or exponentially convergentwithin the ball B(0.68). Moreover, every solution of system (12)satisfies the following estimation

∥x(t, φ,w)∥∞ ≤ 0.68 + 1.25 (∥φ∥∞ − 0.544)+ e−0.0148t , t ≥ 0,

for all φ ∈ C([−6, 0], R3) and w ∈ BF0.5([0, ∞), R). A statetrajectory of the system with w(t) = 0.5 sin 4t is presentedin Fig. 1. Moreover, system (12) without disturbance is globallyexponentially stable by Corollary 7.

5. Conclusion

This note has dealtwith the problemof state bounding for lineartime-varying systems with delay and bounded disturbances. Newdelay-independent conditions in terms of the Metzler matrix havebeen derived for the existence of a ball with the radius explicitlydepends on the upper bound of the disturbances ensuring that allthe state trajectories of the system converge exponentially withinthat ball. An illustrative example has been provided to illustrate theeffectiveness of the obtained result.

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