An improved reciprocally convex inequality and an augmented Lyapunov–Krasovskii functional for stability of linear systems with time-varying delay

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An improved reciprocally convex inequality and an augmented Lyapunov–Krasovskii functional for stability of linear systems with time-varying delay✩ Xian-Ming Zhang a , Qing-Long Han a,∗ , Alexandre Seuret b , Frédéric Gouaisbaut b a

School of Software and Electrical Engineering, Swinburne University of Technology, Melbourne, VIC 3122, Australia

b

CNRS, LAAS, 7 avenue du Colonel Roche, 31077 Toulouse, France

article

info

Article history: Received 26 November 2016 Received in revised form 7 February 2017 Accepted 10 April 2017 Available online xxxx Keywords: Time-delay systems Stability Reciprocally convex inequality Lyapunov–Krasovskii functional

abstract This paper is concerned with stability of a linear system with a time-varying delay. First, an improved reciprocally convex inequality including some existing ones as its special cases is derived. Compared with an extended reciprocally convex inequality recently reported, the improved reciprocally convex inequality can provide a maximum lower bound with less slack matrix variables for some reciprocally convex combinations. Second, an augmented Lyapunov–Krasovskii functional is tailored for the use of a second-order Bessel–Legendre inequality. Third, a stability criterion is derived by employing the proposed reciprocally convex inequality and the augmented Lyapunov–Krasovskii functional. Finally, two wellstudied numerical examples are given to show that the obtained stability criterion can produce a larger upper bound of the time-varying delay than some existing criteria. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction Consider the system with a time-varying delay described by x˙ (t ) = Ax(t ) + Ad x(t − h(t )) x(θ) = φ(θ ), θ ∈ [−hM , 0],



(1)

where x(t ) ∈ Rn is the system state; A and Ad are real n × n constant matrices; h(t ) is the time-varying delay satisfying 0 ≤ h(t ) ≤ hM ,

dm ≤ h˙ (t ) ≤ dM < ∞

(2)

with hM , dm and dM being known constants; and φ(θ ) is an initial condition. The Lyapunov–Krasovskii functional method plus integral inequalities is regarded as a powerful tool for deriving a maximum upper bound hM such that the system (1) can maintain stability for h(t ) ∈ [0, hM ] (Gu, 2013; Gu & Liu, 2009; He, Wang, Lin, & Wu, 2007; Jiang & Han, 2006; Xu, Lam, Zhang, & Zou, 2015;

✩ This work was supported in part by the Australian Research Council Discovery Project under Grant DP160103567. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Keqin Gu under the direction of Editor André L. Tits. ∗ Corresponding author. E-mail addresses: [email protected] (X.-M. Zhang), [email protected] (Q.-L. Han), [email protected] (A. Seuret), [email protected] (F. Gouaisbaut).

http://dx.doi.org/10.1016/j.automatica.2017.04.048 0005-1098/© 2017 Elsevier Ltd. All rights reserved.

Zhang & Han, 2008, 2011). This method has gained increasing attention since the Jensen integral inequality is improved by Wirtinger-based integral inequality (Seuret & Gouaisbaut, 2013), and much effort has been made in seeking less conservative stability criteria for the system (1), e.g. Kim (2016), Zeng, He, Wu, and She (2015b), Zhang and Han (2015a,b) and Zhang, He, Jiang, Wu, and Zeng (2016). It should be mentioned that the Wirtingerbased integral inequality is further improved by Bessel–Legendre inequalities (Seuret & Gouaisbaut, 2015). However, it is found that, if taking some Lyapunov–Krasovskii functional, the stability criterion based on Bessel–Legendre inequalities may be of the same conservatism as the one using the Wirtinger-based integral inequality. To make it clear, we take the proof of Theorem 7 in Seuret and Gouaisbaut (2013) as an example. In what follows, for simplicity of presentation, we denote

 t −h(t )  x(s)  ρ1 (t ) := ds,    h − h( t )  t −t −hMh(t ) M    (t − h(t ) − s)x(s)   ds  ρ2 (t ) := (hM − h(t ))2 t −h  t M  x(s)   ds,  ρ3 (t ) :=   h  t −t h(t ) (t )   (t − s)x(s)   ρ (t ) := ds. 4 h2 ( t ) t −h(t )

(3)

2

X.-M. Zhang et al. / Automatica (

In the proof of Theorem 7 in Seuret and Gouaisbaut (2013), the Lyapunov–Krasovskii functional is chosen as V˜ (xt , x˙ t ) = x˜ T (t )P x˜ (t ) + V˜ 1 (t , xt ) + V2 (t , x˙ t )

(4)

where x˜ (t ) := col{x(t ), h(t )ρ3 (t ), (hM − h(t ))ρ1 (t )}, and V˜ 1 (t , xt ) = V2 (t , x˙ t ) =



t t −h(t )



x (s)Qx(s)ds + T

t t −h M



t

xT (s)Sx(s)ds

(5)

t −hm

 θ

t

x˙ T (s)Rx˙ (s)dsdθ .

(6)

Applying the Wirtinger-based integral inequality, it is proven (Seuret & Gouaisbaut, 2013) that

˙

V˜ (xt , x˙ t ) ≤ ξ1T (t )Φ (h(t ), h˙ (t ))ξ1 (t ),

(7)

where ξ1 (t ) = col{x(t ), x(t − h(t )), x(t − hM ), ρ3 (t ), ρ1 (t )}, and Φ (h(t ), h˙ (t )) is defined in Seuret and Gouaisbaut (2013). However, use Lemma 1 and the reciprocally convex approach (Park, Ko, & Jeong, 2011) to get

˙ V˜ (xt , x˙ t ) ≤ ξ1T (t )Φ (h(t ), h˙ (t ))ξ1 (t ) − ξ2T (t )Ψ ξ2 (t )

(8)

where ξ2 (t ) := col{x(t −h(t ))−x(t −hM )−6ρ1 (t )+12ρ2 (t ), x(t )− S˜ ] ≥ 0. x(t − h(t )) − 6ρ3 (t ) + 12ρ4 (t )} and Ψ := h1 [ 5R M S ˜ T 5R Denote ζ1 (t ) = col{ξ1 (t ), 0, 0}, ζ2 (t ) = col{ξ1 (t ), ρ4 (t ), ρ2 (t )} and Γ = [ 0 I −I 0 −6I 0 12I ]. Then (8) can be rewritten I −I 0 −6I 0 12I 0 as

˙ V˜ (xt , x˙ t ) ≤ ζ1T (t ) diag{Φ (h(t ), h˙ (t )), −I }ζ1 (t ) − ζ2T (t )Γ T Ψ Γ ζ2 (t ).

(9)

Notice that ζ1 (t ) and ζ2 (t ) are linearly independent since ζ1 (t ) does not include the vectors ρ2 (t ) and ρ4 (t ). Thus, the stability criteria derived from (7) and (9) both can be given by the same form as Φ (h(t ), h˙ (t )) < 0 due to Ψ ≥ 0. Therefore, the use of a Bessel–Legendre inequality may not reduce the conservatism of the obtained stability criterion. From the above analysis, it is clear to see that, in order to derive less conservative stability criteria, one should construct a proper Lyapunov–Krasovskii functional such that the corresponding vector ζ1 (t ) is linearly dependent on the vector ζ2 (t ) in (9). To construct such a Lyapunov–Krasovskii functional is the first motivation of the study. On the other hand, the reciprocally convex approach is widely used in the stability analysis of linear systems with time-varying delay. It is because, as stated in Park et al. (2011), the reciprocally convex approach can produce a stability criterion with less decision variables while its conservatism will not be increased. Recently, the reciprocally convex inequality in Park et al. (2011) is extended in Seuret and Gouaisbaut (2016) by introducing four slack matrix variables. Although the extended reciprocally convex inequality is helpful for deriving a less conservative stability condition, the introduction of four slack matrix variables undoubtedly increases the computation complexity of the obtained stability criterion. How to reduce the number of the slack matrix variables of the extended reciprocally convex inequality while the corresponding stability criterion to be of less conservatism is the second motivation of the study. This paper focuses on stability of linear systems with timevarying delay described by (1). First, an improved reciprocally convex inequality is derived, which provides a maximum lower bound for some reciprocally convex combination, while less slack matrix variables are introduced if compared with the extended reciprocally convex inequality (Seuret & Gouaisbaut, 2016). Second, an augmented Lyapunov–Krasovskii functional is tailored for the use of a second-order Bessel–Legendre inequality. On the

)

–

one hand, the terms ρ2 (t ) and ρ4 (t ) appear in the derivative of the Lyapunov–Krasovskii functional, which means that the secondorder Bessel–Legendre inequality can be used to formulate less conservative stability criteria; and on the other hand, the quadratic x˜ T (t )P x˜ (t ) in (4) is deleted. Instead, V˜ 1 (t , xt ) in (5) is augmented so that the relationship between ρj (t ) (j = 1, . . . , 4) and the other vectors is enhanced in the derivative of the Lyapunov–Krasovskii functional. Third, the obtained reciprocally convex inequality and the augmented Lyapunov–Krasovskii functional are employed to derive a stability criterion for the system (1), whose effectiveness is demonstrated through two well-used numerical examples. In this paper, Sym{A} = A+AT . To end this section, we introduce three lemmas, which are useful in the stability analysis. Lemma 1 (Seuret & Gouaisbaut, 2015). For any n × n constant real matrix R > 0, two scalars r1 and r2 with r2 > r1 , and a vector-valued function ω : [r1 , r2 ] → Rn such that the integrations below are well defined, then r2



ω˙ T (s)Rω( ˙ s)ds ≥

r1

1

ζ T (r1 , r2 )R˜ ζ (r1 , r2 )

r2 − r1

(10)

where R˜ := diag {R, 3R, 5R} and ζ (r1 , r2 ) := col{υ0 , υ1 , υ2 } with υ0 := ω(r2 ) − ω(r1 ) and  r2  2  ω(s)ds υ := ω( r ) + ω( r ) −  1 2 1   r2 − r1 r1    r 

υ2 := υ0 −

      

+

6

2

r2 − r1  r1 r2 12

(r2 − r1 )2

ω(s)ds

(11)

(r2 − s)ω(s)ds.

r1

Lemma 2 (Kim, 2016). For a given quadratic function ℓ(s) = a2 s2 + a1 s + a0 , where ai ∈ R (i = 0, 1, 2), if the following inequalities hold (i) ℓ(0) < 0; (ii) ℓ(h) < 0; (iii) − h2 a2 + ℓ(0) < 0

(12)

one has ℓ(s) < 0 for ∀s ∈ [0, h]. Lemma 3 (Zhang & Han, 2013). The system (1) is asymptotically stable if there exists a quadratic Lyapunov–Krasovskii functional ˙ such that for some εi > 0 (i = 1, 2, 3) V (t , φ, φ)

˙ ≤ ε2 ∥φ∥2W ε1 ∥φ(0)∥2 ≤ V (t , φ, φ) ˙ ≤ −ε3 ∥φ(0)∥2 V˙ (t , φ, φ) 0

˙ )∥2 dθ . where ∥φ∥2W = supθ∈[−hM ,0] ∥φ(θ )∥2 + −h ∥φ(θ M 2. Main results 2.1. An improved reciprocally convex inequality Recently, the reciprocally convex inequality proposed in Park et al. (2011) is extended by introducing four slack matrix variables, which is given as follows. Lemma 4 (Seuret & Gouaisbaut, 2016). Let R1 , R2 ∈ Rm×m be real symmetric positive definite matrices and ϖ1 , ϖ2 ∈ Rm and a scalar α ∈ (0, 1). If there exist real symmetric matrices X1 , X2 ∈ Rm×m and real matrices Y1 , Y2 ∈ Rm×m such that



R1 − X1

Y1

Y1T

R2



≥ 0,



R1

Y2

Y2T

R2 − X2

 ≥0

(13)

X.-M. Zhang et al. / Automatica (

the following inequality holds F (α) :=

1

1

ϖ T R1 ϖ1 + ϖ2T R2 ϖ2 ≥ G (X1 , X2 ) (14) α 1 1−α G (X1 , X2 ) := ϖ1T [R1 + (1 − α)X1 ]ϖ1 + ϖ2T (R2 + α X2 )ϖ2 + 2ϖ1T [α Y1 + (1 − α)Y2 ]ϖ2 .

(15)

Lemma 4 presents a general lower bound G (X1 , X2 ) for the reciprocally convex combination F (α) by introducing four slack matrix variables, which bring additional degree of freedom if compared with the one in Park et al. (2011). However, more slack matrix variables usually lead to high computation complexity. The following analysis gives a maximum lower bound for F (α) from the set {G (X1 , X2 )|(X1 , X2 ) satisfies (13)}. Theorem 1. Let R1 , R2 ∈ R be real symmetric positive definite matrices and ϖ1 , ϖ2 ∈ Rm and a scalar α ∈ (0, 1). Then for any Y1 , Y2 ∈ Rm×m , the following inequality holds m×m

)

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a constraint X1 = X2 on X1 and X2 . However, in the case where the dimensions of R1 and R2 are not compatible, one cannot set X1 = X2 ; and in the case where R1 = R2 , the constraint X1 = X2 only makes the lower bound G (X1 , X1 ) deviate away from the maximum G (X10 , X20 ) due to that, in most cases, X10 is not equal to X20 even though one sets Y1 = Y2 . 2.2. An augmented Lyapunov–Krasovskii functional In this section, we introduce a Lyapunov–Krasovskii functional candidate as V (t , xt , x˙ t ) = V1 (t , xt ) + hM V2 (t , x˙ t )

V1 (t , xt ) :=

t −h(t )



η2T (t , s)Q2 η2 (t , s)ds

(21)

where Q1 > 0, Q2 > 0 and R > 0 are to be determined, and

+ 2ϖ [α Y1 + (1 − α)Y2 ]ϖ2 . T 1

(16)

Proof. Since R1 > 0 and R2 > 0, the matrix inequalities in (13) are equivalent to, respectively, Y2T R1−1 Y2

−1 T

η1T (t , s)Q1 η1 (t , s)ds

t −h(t )

t −h M

+ ϖ2T [R2 + α(R2 − Y2T R1−1 Y2 )]ϖ2

R2 − X2 −

t



+

−1

R1 − X1 − Y1 R2 Y1 ≥ 0,

(20)

where xt := x(t + θ ), θ ∈ [−hM , 0]; V2 (t , x˙ t ) is defined in (6),

F (α) ≥ ϖ1T [R1 + (1 − α)(R1 − Y1 R2 Y1T )]ϖ1

−1 T

3

≥ 0.

(17)

Y2T R1−1 Y2 .

η1 (t , s) := col{˙x(s), x(s), η0 (t ),



η2 (t , s) := col{˙x(s), x(s), η0 (t ),



s

x(θ )dθ }

(22)

x(θ )dθ }

(23)

t −h(t ) s t −h M

η0 (t ) := col {x(t ), x(t − h(t )), x(t − hM )} .

Denote X10 = R1 − Y1 R2 Y1 and X20 = R2 − Then it follows from (17) that X10 ≥ X1 and X20 ≥ X2 , which leads to G (X10 , X20 ) ≥ G (X1 , X2 ), where G (X1 , X2 ) is defined in (15). Since (X10 , X20 ) satisfies (13), by Lemma 4, one obtains

It is not difficult to verify that there exist two constants c1 > 0 and c2 > 0 such that

F (α) ≥ G (X10 , X20 ) ≥ G (X1 , X2 )

In fact, denote ϵ0 = min{λmin (Q1 ), λmin (Q2 )}. Then

which completes the proof.

(18)



Remark 2. In Zhang et al. (2016), some specific reciprocally convex inequalities are derived based on Wirtinger-based integral inequality (Seuret & Gouaisbaut, 2013) and the integral inequality (10), respectively, see Lemmas 4 and 6 therein. For comparison, we rewrite the results in Lemmas 4 and 6 in Zhang et al. (2016) in a general form as

  R + (1 − α)T1 β β1T Rβ1 + β2T Rβ2 ≥ 1 β2 ST α 1−α   β × 1 β2 1

S



(19)

where β1 and β2 are two vectors; R > 0, T1 = R − SR−1 S T and T2 = R − S T R−1 S. One can see that, if taking ϖi = βi , Ri = R (i = 1, 2) and Y1 = Y2 = S, the inequality (16) immediately reduces to (19), which means that Lemmas 4 and 6 in Zhang et al. (2016) are special cases of Theorem 1. Moreover, the proof of Theorem 1 is much simpler than that of Lemma 4 or Lemma 6 in Zhang et al. (2016). Remark 3. In Seuret and Gouaisbaut (2016), it is suggested to reduce the number of decision variables in Lemma 4 by imposing

ϵ0 x (t )x(t )ds + T

t −h(t )

(24)

t −h(t )



ϵ0 xT (t )x(t )ds

t −h M

= hM ϵ0 xT (t )x(t ) , c1 xT (t )x(t ).

(25)

On the other hand, denote ϵ1 = max{λmax (Q1 ), λmax (Q2 )}. Then V1 (t , xt ) ≤ ϵ1

t



  ∥˙x(s)∥22 + ∥x(s)∥22 + ∥η0 (t )∥22 ds t −hM

t

+ ϵ1



+ ϵ1



s



t −h(t )

t −h(t )

t −h(t )



t −hM

≤ ϵ1



∥xT (θ )∥2 dθ



s

∥xT (θ )∥2 dθ t −hM

s

t −h(t )



∥x(θ )∥2 dθ ds

s

∥x(θ )∥2 dθ ds t −h M

t

  ∥˙x(s)∥22 + 3∥x(s)∥22 ds + ϵ1 hM ∥xt (0)∥22 t −hM

+ ϵ1 hM

R + α T2

t



V1 (t , xt ) ≥

Remark 1. Compared with Lemma 4, Theorem 1 provides a maximum lower bound G (X10 , X20 ) for F (α). It is worth pointing out that the slack matrix variables X1 and X2 in (13) are removed from Theorem 1. Furthermore, if one sets Y1 = Y2 = S, Theorem 1 reduces to Theorem 3 in Zhang and Han (in press); and it is easy to show that G (X10 , X20 ) is greater than ϖ1T R1 ϖ1 + 2ϖ1T S ϖ2 + ϖ2T R2 ϖ2 , which is proposed in Park et al. (2011).

1

c1 ∥xt (0)∥22 ≤ V (t , xt , x˙ t ) ≤ c2 ∥xt ∥2W .





t −h(t )

t



t −h(t )



t

+ t −hM

t

∥x(θ )∥22 dθ ds t −hM

 ∥x(θ )∥22 dθ ds

t −hM

≤ ϵ1 hM (h2M + 4)

sup

θ ∈[−hM ,0]

V2 (t , x˙ t ) ≤ hM λmax (R)



∥xt (θ )∥22 + ϵ1



t

∥˙x(s)∥22 ds, t −h M

t

∥˙x(s)∥22 ds t −hM

which leads to V (t , xt , x˙ t ) ≤ c2 ∥xt ∥2W , where c2 := max{ϵ1 hM (h2M + 4), h2M λmax (R) + ϵ1 }, and ∥φ∥22 , φ T φ . Compared with some existing Lyapunov–Krasovskii functionals, e.g. (4) and those in He, Wang, Lin, and Wu (2005), Kim (2016)

4

X.-M. Zhang et al. / Automatica (

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and Seuret and Gouaisbaut (2013), V (t , xt ) in (20) has the following characteristics:

Proof. Taking the time derivative of V (t , xt ) in (20) along with the trajectory of the system (1) yields

(i) The quadratic term, i.e. xT (t )Px(t ) or x˜ T (t )P x˜ (t ) in (4), is deleted. Instead, an augmented integral term V1 (t , xt ) is introduced. As a result, the system states x(s), x(t ), x(t − d(t )) and x(t − hM ) are closely coupled by Q1 and Q2 . Such coupling can enhance the relationship between x(t ) and the other delayed state vectors in the derivative of V (t , xt ); and (ii) Taking the time-derivative of V (t , xt ) yields two important integrals ρ2 (t ) and ρ4 (t ). That is, the corresponding vector ζ1 (t ) in (9) is the same as ζ2 (t ), which means that one can employ the integral inequality (10) to derive less conservative stability conditions.

V˙ (t , xt ) = V˙ 1 (t , xt ) + hM V˙ 2 (t , xt )

2.3. Stability analysis In this section, we are to establish a stability criterion for the system (1) based on Theorem 1 and the Lyapunov–Krasovskii functional (20). Proposition 1. For given scalars hM , dm and dM (<1), the system (1) is asymptotically stable if there exist real matrices Q1 > 0, Q2 > 0, R > 0, Y1 and Y2 with appropriate dimensions such that for d ∈ {dm , dM }



Υ1 (0, d) Y 2 Γ2

 Υ 1 ( hM , d ) Y1T Γ1

Γ2T Y2 < 0, −R˜  2 −hM G0 (d) + Υ1 (0, d) Y 2 Γ2

 T

Γ1T Y1 <0 −R˜  Γ2T Y2T <0 −R˜



(26) (27)

where R˜ = diag {R, 3R, 5R} and

Υ1 (h(t ), h˙ (t )) := [C11 + h(t )C12 ]T Q1 [C11 + h(t )C12 ]

+ h2M C0T RC0 − C5T Q2 C5 − (1 − h˙ (t ))C2T Q1 C2 − (2 − α)Γ1T R˜ Γ1 + (1 − h˙ (t ))[C41 + (hM − h(t ))C42 ]T × Q2 [C41 + (hM − h(t ))C42 ]   + Sym D1T Q1 [C30 + h(t )C31 + h2 (t )C32 ]   + Sym D2T Q2 [C60 + (hM − h(t ))C61 + (hM − h(t ))2 C62 ]   − (1 + α)Γ2T R˜ Γ2 − Sym Γ1T [α Y1 + (1 − α)Y2 ]Γ2 (28) T T G0 (h˙ (t )) := C12 Q1 C12 + (1 − h˙ (t ))C42 Q2 C42   + Sym D1T Q1 C32 + D2T Q2 C62 (29)

where V˙ 1 (t , xt ) = η1T (t , t )Q1 η1 (t , t ) − η2T (t , t − hM )Q2 η2 (t , t − hM )

− (1 − h˙ (t ))η1T (t , t − h(t ))Q1 η1 (t , t − h(t )) + (1 − h˙ (t ))η2T (t , t − h(t ))Q2 η2 (t , t − h(t ))  t ∂η1 (t , s) ds 2η1T (t , s)Q1 + ∂t t −h(t )  t −h(t ) ∂η2 (t , s) 2η2T (t , s)Q2 ds + ∂t t −hM  t x˙ T (s)Rx˙ (s)ds. hM V˙ 2 (t , xt ) = h2M x˙ T (t )Rx˙ (t ) − hM

C12 := col{0, 0, 0, 0, 0, e6 }

C2 := col{e8 , e2 , e1 , e2 , e3 , 0}, C30 := col{e1 − e2 , 0, 0, 0, 0, 0} C31 := col{0, e6 , e1 , e2 , e3 , 0},

C32 := col{0, 0, 0, 0, 0, e7 }

C41 := col{e8 , e2 , e1 , e2 , e3 , 0},

C42 := col{0, 0, 0, 0, 0, e4 }

C5 := col{e9 , e3 , e1 , e2 , e3 , 0}, C60 := col{e2 − e3 , 0, 0, 0, 0, 0}

ξ (t ) := col{x(t ), x(t − h(t )), x(t − hM ), ρ1 (t ), ρ2 (t ), ρ3 (t ), ρ4 (t ), x˙ (t − h(t )), x˙ (t − hM )} (33) where ρj (t ) (j = 1, 2, 3, 4) are defined in (3). Then one has x˙ (t ) = C0 ξ (t ), and

η1 (t , t ) = (C11 + h(t )C12 )ξ (t ), η1 (t , t − h(t )) = C2 ξ (t ), η2 (t , t − h(t )) = [C41 + (hM − h(t ))C42 ]ξ (t ), η2 (t , t − hM ) = C5 ξ (t ),  t ∂η1 (t , s) 2η1T (t , s)Q1 ds ∂t t −h(t )   = 2ξ T (t )D1T Q1 C30 + h(t )C31 + h2 (t )C32 ξ (t ), (34)  t −h(t ) ∂η2 (t , s) 2η2T (t , s)Q2 ds = 2ξ T (t )D2T Q2 ∂t t −h M   × C60 + (hM − h(t ))C61 + (hM − h(t ))2 C62 ξ (t ). (35) t Denote I (t ) := hM t −h x˙ T (s)Rx˙ (s)ds. Then M  t  t −h(t ) T x˙ (s)Rx˙ (s)ds + hM x˙ T (s)Rx˙ (s)ds. I ( t ) = hM t −h(t )

t −hM

Applying Lemma 1 yields



t −h(t )

hM

x˙ T (s)Rx˙ (s)ds ≥

t −hM

 hM

t

t −h(t )

x˙ T (s)Rx˙ (s)ds ≥

1

α

(Γ1 ξ (t ))T R˜ (Γ1 ξ (t )) 1

1−α

(Γ2 ξ (t ))T R˜ (Γ2 ξ (t ))

˜ where α = (hM − h(t ))/hM . Thus, apply (16) with R1 = R2 = R, ϖ1 = Γ1 ξ (t ) and ϖ2 = Γ2 ξ (t ) to obtain 

˜ −1 Y1T Γ1 I (t ) ≥ ξ T (t ) Υ0 − (1 − α)Γ1T Y1 R  − α Γ2T Y2T R˜ −1 Y2 Γ2 ξ (t )

D1 := col{0, 0, C0 , (1 − h˙ (t ))e8 , e9 , (h˙ (t ) − 1)e2 } D2 := col{0, 0, C0 , (1 − h˙ (t ))e8 , e9 , −e3 }

To sum up, one has that

Γ1 := col{e2 − e3 , e2 + e3 − 2e4 , e2 − e3 − 6e4 + 12e5 } Γ2 := col{e1 − e2 , e1 + e2 − 2e6 , e1 − e2 − 6e6 + 12e7 }

V˙ (t , xt ) ≤ ξ T (t )[Υ1 (h(t ), h˙ (t )) + Υ2 (h(t ))]ξ (t )

C62 := col{0, 0, 0, 0, 0, e5 }

with ei (i = 1, 2, . . . , 9) being the ith n × 9n block-row vectors of the 9n × 9n identity matrix.

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For simplicity, denote

Υ0 := (2 − α)Γ1T R˜ Γ1 + (1 + α)Γ2T R˜ Γ2   + Sym Γ1T [α Y1 + (1 − α)Y2 ]Γ2 .

C61 := col{0, e4 , e1 , e2 , e3 , 0},

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t −hM

where C0 := Ae1 + Ad e2 , α = (hM − h(t ))/hM and C11 := col{C0 , e1 , e1 , e2 , e3 , 0},

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where Υ1 (h(t ), h˙ (t )) is defined in (28), and

Υ2 (h(t )) := (1 − α)Γ1T Y1 R˜ −1 Y1T Γ1 + α Γ2T Y2T R˜ −1 Y2 Γ2 .

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X.-M. Zhang et al. / Automatica ( Table 1 The maximum admissible upper bound hM for d = −dm = dM for Example 1. Method

d

Zhang et al. (2016) Kim (2016) Zeng et al. (2015a) Proposition 1

–

5

Table 2 The maximum admissible upper bound hM for d = −dm = dM for Example 2.

NoDVs

0.1

0.5

0.8

4.714

2.608

2.375

100

4.753 4.788 4.910

2.429 3.055 3.233

2.183 2.615 2.789

116 282 231

Note that

Υ1 (h(t ), h˙ (t )) + Υ2 (h(t )) = h2 (t )G0 (h˙ (t )) + h(t )G1 (h˙ (t )) + G2 (h˙ (t )) where G0 (h˙ (t )) is defined in (29); G1 (h˙ (t )) and G2 (h˙ (t )) are some proper real symmetric matrices irrespective of h(t ). Let us consider the quadratic function χ T [Υ1 (h(t ), h˙ (t )) + Υ2 (h(t ))]χ as

χ T [Υ1 (h(t ), h˙ (t )) + Υ2 (h(t ))]χ = a2 h2 (t ) + a1 h(t ) + a0

)

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where a0 = χ T G2 (h˙ (t ))χ , a1 = χ T G1 (h˙ (t ))χ and a2 = χ T G0 (h˙ (t ))χ with χ ∈ R9n . If (26) and (27) are satisfied, applying Lemma 2 and the Schur complement yields χ T [Υ1 (h(t ), h˙ (t )) + Υ2 (h(t ))]χ < 0 for h(t ) ∈ [0, hM ] and h˙ (t ) ∈ [dm , dM ]. Let χ = ξ (t ). Then one has V˙ (t , xt ) ≤ ξ T (t )[Υ1 (h(t ), h˙ (t )) + Υ2 (h(t ))]ξ (t ) < 0 for h(t ) ∈ [0, hM ] and h˙ (t ) ∈ [dm , dM ]. Thus, by Lemma 3, one can draw a conclusion that the system (1) is asymptotically stable. 

Method

d

Seuret (2013) Kwon et al. (2014) Proposition 1

0.1

0.2

0.5

0.8

6.590 7.125 7.230

3.672 4.413 4.556

1.411 2.243 2.509

1.275 1.662 1.940

For comparison with some existing stability criteria, we calculate the maximum admissible upper bound hM for different d = −dm = dM . For d ∈ {0.1, 0.5, 0.8}, applying the approaches in Kim (2016), Zeng et al. (2015a), Zhang et al. (2016) and Proposition 1, the obtained results are given in Table 1. Moreover, the number of decision variables (NoDVs) involved in solving the corresponding linear matrix inequalities is also listed in Table 1. From this table, one can see clearly that Proposition 1 delivers some larger upper bounds hM for the time-varying delay h(t ) than those by Kim (2016), Zeng et al. (2015a), Zhang et al. (2016), and the number of decision variables is also smaller than that in Zeng et al. (2015a). Although Proposition 1 requires more decision variables than (Kim, 2016; Zhang et al., 2016), for d = 0.5, Proposition 1 improves the upper bound hM = 2.608 (Zhang et al., 2016) by 23.98%, and the upper bound hM = 2.429 (Kim, 2016) by 33.1%. Example 2. Consider the system (1) with

 A=

0 −1



1 , −2

 Ad =

0 −1



0 1

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and the delay h(t ) is time-varying satisfying (2).

Remark 4. Proposition 1 presents a stability criterion based on the augmented Lyapunov–Krasovskii functional. From the proof, it is clear that in the estimation of V˙ (t , xt ), Theorem 1 and Lemmas 1 and 2 play a key role in deriving a tight upper bound for V˙ (t , xt ). For the system (1), Proposition 1 requires 54.5n2 + 6.5n decision variables. This number is less than 65n2 + 11n of Theorem 1 in Zeng, He, Wu, and She (2015a) while more than 23n2 + 4n of Theorem 2 in Zhang et al. (2016) and 27n2 + 4n of Theorem 1 in Kim (2016). However, simulation results in the next section show that Proposition 1 can produce less conservative results than some existing stability criteria, including those mentioned above.

For this example, in Kwon, Park, Park, Lee, and Cha (2014) and Seuret and Gouaisbaut (2013), the maximum admissible upper bound hM of the time-varying delay h(t ) is calculated for d = −dm = dM ∈ {0.1, 0.2, 0.5, 0.8} and the obtained results are listed in Table 2. However, applying Proposition 1 yields some larger upper bounds hM , which are also given in this table. It is clear to see that for d = 0.8, the maximum upper bound hM is improved by 16.73% and 52.16%, respectively, if compared with the ones in Kwon et al. (2014) and Seuret and Gouaisbaut (2013).

Remark 5. It is worth pointing out that Proposition 1 is infeasible for dM ≥ 1 due to the diagonal entry (1 − h˙ (t ))eT8 (Q2 − Q1 )e8 in Υ1 (h(t ), h˙ (t )). In the case where dM ≥ 1, similar to the proof of Proposition 1, one can establish a stability criterion for the system (1) if η1 (t , s) and η2 (t , s) in (22) and (23) are replaced with η˜ 1 (t , s) and η˜ 2 (t , s), respectively, as

The problem of stability of linear systems with time-varying delay has been addressed. By introducing an improved reciprocally convex inequality and an augmented Lyapunov–Krasovskii functional, a stability criterion has been derived for the system under study, whose effectiveness has been demonstrated through two well-used numerical examples.

   s η˜ 1 (t , s) := col x(s), x(t ), x(t − hM ), x(θ )dθ t −h(t )    s η˜ 2 (t , s) := col x(s), x(t ), x(t − hM ), x(θ )dθ .

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t −hM

Due to page limitation, this result is omitted. 3. Numerical examples Example 1. Consider the system (1) with A=

 −2 0



0 , −0.9



−1 Ad = −1



0 −1

and the delay h(t ) is time-varying satisfying (2).

4. Conclusion

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