Further refinements in stability conditions for time-varying delay systems

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Further refinements in stability conditions for time-varying delay systems Fúlvia S.S. de Oliveira a,∗, Fernando O. Souza b a

Graduate Program in Electrical Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG 31270-010, Brazil b Department of Electronics Engineering, Federal University of Minas Gerais, Av. Antônio Carlos 6627, Belo Horizonte, MG 31270-010, Brazil

a r t i c l e

i n f o

Article history: Received 27 March 2019 Revised 11 October 2019 Accepted 21 October 2019 Available online xxx Keywords: Stability analysis Time-varying delay Lyapunov-Krasovskii functional Linear matrix inequalities

a b s t r a c t This paper addresses the problem of assessing the stability of linear time-invariant (LTI) systems with time-varying delay. The first contribution is a new stability criterion specified as a negativity condition for a quadratic function parameterized by the delay. This result mainly follows from an augmented affine parameter-dependent Lyapunov-Krasovskii functional, which, in turn, takes advantages of convexity properties. Then, as a second contribution, we invoke a result from robust control literature to show how the proposed stability condition can be checked exactly in terms of linear matrix inequality (LMI) conditions. The improvements obtained by the proposed refinements are illustrated via numerical examples drawn from the literature. © 2019 Elsevier Inc. All rights reserved.

1. Introduction Over the past years, stability analysis of time-delay systems has been a very active research field in control theory, due to the fact that time-delay is a common phenomenon in a variety of control systems and it can be source of performance degradation and instability [1,2]. Moreover, time-delay systems belong to the class of infinite-dimensional differential equations, which make its stability determination a very complex problem. Namely, the main goal of this paper is to characterize whether the origin is an asymptotically stable equilibrium point of the linear state-delayed system given by



x˙ (t ) = Ax(t ) + Ad x(t − h(t )) x(θ ) = ϕ (θ ), θ : [−hM , 0] → Rn

(1)

where x(t ) ∈ Rn , A, Ad ∈ Rn×n , ϕ (θ ) is an initial condition, and h(t) is a continuous function used to describe the timevarying delay satisfying

0 ≤ h(t ) ≤ hM ,

d1 ≤ h˙ (t ) ≤ d2 < 1,

(2)

with hM , d1 , and d2 being known constant scalars. There are a number of methods for analyzing stability of time-varying delay systems, both in frequency and time domains. For a survey on these methods the readers are referred to [1,2]. Concerning the frequency domain methods, an ∗

Corresponding author. E-mail address: [email protected] (F.S.S. de Oliveira).

https://doi.org/10.1016/j.amc.2019.124866 0 096-30 03/© 2019 Elsevier Inc. All rights reserved.

Please cite this article as: F.S.S. de Oliveira and F.O. Souza, Further refinements in stability conditions for time-varying delay systems, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124866

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Fig. 1. Graphical interpretation of Lemma 1. The highlighted points correspond to the points tested by conditions (i), (ii), and (iii).

effective approach is the integral quadratic constraints (IQC) [3–5]. In IQC framework the delay system is rewritten as an interconnection of a delay free system and a delay operator. Then, the basic idea is to replace the delay operator with an IQC characterization, which contains all possible solutions of the delay operator. The infinite-dimensional frequency-dependent stability conditions can be formulated as finite-dimensional frequency-independent linear matrix inequalities (LMI) conditions by the use of the Kalman-Yakubovich-Popov lemma [6]. Likewise, the direct Lyapunov-Krasovskii method allied with quadratic functionals is a very effective methodology to investigate the stability, control, and state observer of a large class of time-delay systems [7–19]. This method usually leads to conditions formulated in terms of LMIs, which can be efficiently solved by convex optimization techniques. The present paper is based on the latter methodology. In the literature, when considering the Lyapunov-Krasovskii method combined with LMI-based approaches, several strategies have been proposed to reduce the conservatism, such as: discretization/partition delay method [1,20,21], new functional choices [22–28], free-weighting matrices [29–33], and improved quadratic integral inequalities [34–39]. This last approach is regarded as a convenient manner to relax the stability criteria due to the low complexity of the resulting LMIs when compared to the discretization methods. A comprehensive study on the effect of bounding inequalities on the conservatism of criteria can be found in [40]. It is interesting to point out that similar methodologies are also employed for stability analysis of sampled-data systems, see [41–43] for some results in this direction. Recently, by employing improved integral inequalities, in [44,45] were presented less conservative stability criteria expressed as negativity conditions for quadratic functions within a finite interval of the form

F (h(t )) = h2 (t )A2 + h(t )A1 + A0 ≺ 0, ∀h(t ) ∈ [0, hM ], where A2 , A1 , and A0 are real matrices of appropriate dimensions and h(t) is the time-delay. Handling this type of condition is difficult because F (h(t )) is non-convex with respect to h(t) on [0, hM ] if A2 ≺ 0. Since the pointwise evaluation of a quadratic function within the finite interval is not practical, in [44,45] the authors make use of sufficient conditions to realize such evaluation. Thus, these methods may be improved. In this paper, based on a new augmented affine parameter-dependent Lyapunov-Krasovskii functional it is proposed a less conservative stability criterion in terms of a negativity condition for a quadratic function within a finite interval. In addition, it is shown how to attest such negativity condition via a finite dimensional convex optimization problem that can be checked exactly in terms of LMIs. Applications and the effectiveness of the proposed methods are illustrated via numerical examples drawn from the literature. 2. Notation and auxiliary lemmas First we introduce some basic notations. Let Rm×n be the set of m × n real matrices and Sn (Sn+ ) the set of n × n real symmetric (positive definite) matrices. ∗ refers to symmetric terms in a symmetric matrix. For a matrix M denote its transpose by MT . For a symmetric matrix M, M0 (M≺0) means that M is positive (negative) definite. Shorthand notations, for a matrix M denote M + MT by sym{M}, for a vector pair (x, y) the notation col{x, y} denotes a column vector with x stacked on top of y, and for a matrix pair (X, Y) denote a diagonal matrix whose elements are X and Y by diag{X, Y}. In the following, we present some lemmas that are essential in proving the results of this paper. The next lemma presents some simple conditions for ensuring the negativity of a quadratic polynomial function within a finite interval. Lemma 1 [45]. Let f ( h ¯ ) = a2 h ¯ 2 + a1 h ¯ + a0 , where a2 , a1 , a0 ∈ R. If

( i ) f ( 0 ) < 0, then f() < 0, ∀

(ii ) f ( h¯ M ) < 0,

(iii ) f (0 ) − h¯ 2M a2 < 0,

 ∈ [ 0, M ] .

Fig. 1 presents a graphical interpretation of Lemma 1. The highlighted points in the graphs correspond to the points tested by conditions (i), (ii), and (iii). One can note that if a2 > 0, f() is a convex function and the conditions (i) and (ii) are necessary and sufficient to ensure f() < 0 within the interval [0, M ] (see Fig. 1a). On the other hand, if f() is concave the conditions (i) and (ii) are no longer sufficient, see graphs b) and c) in Fig. 1. This explains the need of condition (iii), Please cite this article as: F.S.S. de Oliveira and F.O. Souza, Further refinements in stability conditions for time-varying delay systems, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124866

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3

which, however, is not necessary. Condition (iii) implies that the tangent line to the curve f() at the point M should be negative at h ¯ = 0. Thus it may be quite conservative, especially when the tangent line has a slope that is far from zero. The graphs b) and c) in Fig. 1 illustrate the conservativeness of condition (iii). Particularly, for the function depicted in Fig. 1c, the condition (iii) does not hold and the Lemma 1 fails, which shows how the conservatism of condition (iii) may lead to erroneous conclusion. A contribution of this paper is to present how the negativity of a quadratic function within a closed interval can be tested non conservatively in terms of LMIs. To this end, we consider the following lemma borrowed from the robust control literature. Lemma 2 [46,47]. Let  ∈ Sm , J, C ∈ Rk×m . The following statements are equivalent. (i) ζ T ζ < 0, ∀ ζ = 0 ∈ Rm which satisfy (J − δC )ζ = 0, for some real scalar δ such that |δ | ≤ 1. (ii) There exist D ∈ Sk+ and skew-symmetric matrix G ∈ Rk×k such that

 T 

C ≺ J

−D GT

G D

 

C . J

The proof of Lemma 2 can be found, for example, in [47]. Additionally, the proof of our main result requires the following two remarkable lemmas, which play a crucial role in obtaining less conservative stability conditions for systems with time-varying delay. Lemma 3 (Second order Bessel-Legendre inequality [36]). For any R ∈ Sn+ , any differentiable function x in [a, b] → Rn , the following inequality holds



b a

1 T diag(R, 3R, 5R ), b−a

x˙ T (u )Rx˙ (u )du ≥

where  = col{1 , 2 , 3 }, with 1 = x(b) − x(a ) and

2 = x ( b ) + x ( a ) − 3 = 1 −

6 b−a

 a

2 b−a b



b a

x(u )du,

x(u )du +

12 ( b − a )2

 a

b

(b − u )x(u )du.

m and a scalar α ∈ (0, 1). If there exist matrices Lemma 4 (Reciprocally Convex lemma [44]). Let R1 , R2 ∈ Sm + ; σ1 , σ2 ∈ R m×m m X1 , X2 ∈ S and Y1 , Y2 ∈ R such that



R1 − X1 ∗



Y1  0, R2



R1 ∗



Y2  0, R2 − X2

(3)

then the following inequality holds

1

α

σ1T R1 σ1 +

1 σ T R2 σ2  σ1T [R1 + (1 − α )X1 ]σ1 + σ2T (R2 + α X2 )σ2 + 2σ1T [αY1 + (1 − α )Y2 ]σ2 . 1−α 2

3. Main results In a similar manner as in the recent papers [44,45] the stability criterion provided in this paper in its primary form is expressed in terms of the negativity of a quadratic function parameterized by the delay. Then in order to avoid the inherent conservatism of Lemma 1, the first contribution of this paper is to show how one can check the quadratic function negativity within a finite interval exactly in terms of LMIs. This technical result is stated in Lemma 6 and the following result is used in its proof.







Lemma 5. Given J = ( h −I p ∈ R p×2 p and C = ( h ¯ M /2 )I p ¯ M /2 )I p and Z2 defined in the following are equal.

Z1 := Z2 :=

 



0 p ∈ R p×2 p , with

M being a known scalar, the sets Z1

ζ ∈ R2 p : (J − δC )ζ = 0, some δ ∈ [−1, 1] ,

ζ ∈ R2 p : ζ = [Ip h¯ Ip ]T ξ , h¯ ∈ [0, h¯ M ], ξ ∈ R p .

Proof. Z2 ⊆ Z1 : Let ζ ∈ Z2 and note that δ ∈ [−1, 1] can be written as a function of

[J − ( 1 − 2 h ¯ /h ¯ M )C]ζ = =



 

( h¯ M /2 )Ip 

0

= h ¯ Ip





−I p − (1 − 2 h ¯ M /2 )I p ¯ /h ¯ M ) (h



−I p + 2 h ¯ M /2 )I p ¯ /h ¯ M (h



−I p I p

h ¯ Ip

T

0p

 ζ

 0p ζ

 ∈ [0, M ] as δ = 1 − 2 h¯ / h¯ M then

ξ = 0.

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Z1 ⊆ Z2 : Let ζ ∈ Z1 and rewrite ζ = col{ξ , ν}, with ξ , ν ∈ R p . Thus







−I p − δ ( h ¯ M /2 )I p

( h¯ M /2 )Ip 

0p



   ξ = 0, some δ ∈ [−1, 1], ν

Ip ξ. implies that ζ = ( h¯ M /2 )(1 − δ )I p Then the claim holds because the image of the function f (δ ) = ( h ¯ M /2 )(1 − δ ) for all δ ∈ [−1, 1] lies in the domain of .  Lemma 6. Let A2 , A1 , A0 ∈ S p and ξ ∈ R p . Then the inequality

 ξ T h¯ 2 A2 + h¯ A1 + A0 ξ < 0

holds for all



A0 1 A 2 1

where

 ∈ [0, M ] if and only if there exist D ∈   T    1 A 2 1





Proof. Let ζ = I p

ξT

Ip h ¯ Ip

T 

−D GT

C J

≺

A2

C = (h ¯ M /2 )I p



(4)

0p h ¯ Ip

A0 1 A 2 1

T



G D

p S+

and skew-symmetric matrix G ∈ R

p×p

such that

C , J

(5)





and J = ( h ¯ M /2 )I p

−I p .

(6)

ξ and rewrite (4) as

1 A 2 1

A2



Ip h ¯ Ip



ξ < 0.

Further, from Lemma 5 it follows that all ζ = 0 that satisfies (J − δC )ζ = 0, for some δ ∈ [−1, 1], is of the form ζ =



Ip

h ¯ Ip

T

ξ , with C and J defined in (6). Then an application of Lemma 2 concludes the proof. 

To assess the stability of systems with time-varying delay given by (1), we propose the use of the following augmented affine parameter-dependent Lyapunov-Krasovskii functional candidate inspired by Lee and Park [25,44]:

V (xt , x˙ t ) = V1 (xt ) + V2 (xt , x˙ t ) + hMV3 (x˙ t ),

(7)

where xt = x(t + θ ), θ ∈ [−hM , 0] and

V1 (xt ) = η1T (t )S1 (t )η1 (t ) + η2T (t )S2 (t )η2 (t ),  t  t −h(t ) V2 (xt , x˙ t ) = η3T (t, s )Q1 η3 (t, s )ds + η4T (t, s )Q2 η4 (t, s )ds, V3 (x˙ t ) =



t −h(t )

t −h(t )

t−hM

t−hM

(hM − t + s )x˙ T (s )R1 x˙ (s )ds +



t

t −h(t )

(hM − t + s )x˙ T (s )R2 x˙ (s )ds,

with S1 (t ) = h(t )S11 + S12 , S2 (t ) = (hM − h(t ))S21 + S22 ,

  η0T (t ) = xT (t ) xT (t − h(t )) xT (t − hM ) ,   t T η1T (t ) = xT (t ) xT (t − h(t )) , t −h(t ) x (s )ds   t −h ( t ) T η2T (t ) = xT (t − h(t )) xT (t − hM ) , t−hM x (s )ds   t T T T T T η3 (t, s ) = x˙ (s ) x (s ) η0 (t ) , s x ( θ )d θ   t −h(t ) T T T T T η4 (t, s ) = x˙ (s ) x (s ) η0 (t ) x ( θ )d θ . s

The differences between this functional and the one in [44] are: (i ) the inclusion of the parameter dependent term V1 (xt ), (ii ) the change on the limits of integration of the integral terms in η3 (t, s) and η4 (t, s), and (iii ) the integration over [0, hM ] in the term V3 (x˙ t ) is split in two integrals. As will be illustrated in Section 4 these changes are important in reducing conservatism. An important feature that the proposed functional incorporates to the one provided in [44] is that the resulting stability condition takes more advantages of convexity properties in terms of the parameters h(t) and h˙ (t ). Specifically, the idea behind the term V3 (x˙ t ) is to derive conditions more suitable with the convex combination in Lemma 4. In light of the Lyapunov-Krasovskii functional candidate in (7) the main result of this paper is stated in the next theorem. Theorem 1. For given scalars d1 , d2 , and hM , system (1) is asymptotically stable if there exist matrices S11 ∈ S3n , S12 ∈ S3+n , S21 ∈ S3n , S22 ∈ S3+n , Qi ∈ S6+n , Ri ∈ Sn+ , Xi ∈ S3n , Yi ∈ R3n×3n , i = 1, 2, such that the following conditions hold for d j=1,2 , and ∀ h(t) ∈ [0, hM ].

hM S11 + S12  0, hM S21 + S22  0,

(8)

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R˜1 − X1 ∗

Y1  0, R˜2



R˜1 ∗

5



Y2  0, R˜2 − X2

(9)

h2 (t ) 2 (d j ) + h(t ) 1 (d j ) + 0 (d j ) ≺ 0,

(10)

where R˜i = diag{Ri , 3Ri , 5Ri } for i = 1, 2 and T

0 (h˙ (t )) = sym{D1T S12 C11 } + h˙ (t )C11 S11C11 + sym{D2T (hM S21 + S22 )C21 } T T T − h˙ (t )C21 S21C21 + C3T Q1 C3 − (1 − h˙ (t ))C41 Q1 C41 + sym{C50 Q1 D3 } T + (1 − h˙ (t ))C6T Q2 C6 − C71 Q2 C71 + sym{(C80 + hM C81 + h2M C82 )T Q2 D4 }

+ h2M C0T R2 C0 + h2M (1 − h˙ (t ))eT8 (R1 − R2 )e8 − 1T R˜1 1 − 2T (R˜2 + X2 )2 − sym{1T Y1 2 },

(11)

T

1 (h˙ (t )) = sym{D1T S12 C12 + D2T (hM S21 + S22 )C22 + h˙ (t )C11 S11 C12 T T T − D2T S21 C21 − h˙ (t )C21 S21 C22 + D1T S11 C11 + C51 Q1 D3 − (1 − h˙ (t ))C41 Q1 C42 T T − C71 Q2 C72 − (C81 + 2hM C82 )Q2 D4 } − (1 − h˙ (t ))hM eT8 (R1 − R2 )e8

− (1/hM )1T X1 1 + (1/hM )2T X2 2 + sym{(1/hM )1T (Y1 − Y2 )2 },

(12)

2 (h˙ (t )) = sym{D1T S11 C12 − D2T S21 C22 + D3T Q1 C52 + D4T Q2 C82 } T T T T + h˙ (t )C12 S11 C12 − h˙ (t )C22 S21 C22 − (1 − h˙ (t ))C42 Q1 C42 − C72 Q2 C72 ,

(13)

with,

1 = col{e2 − e3 , e2 + e3 − 2e4 , e2 − e3 − 6e4 + 12e5 }, 2 = col{e1 − e2 , e2 + e1 − 2e6 , e1 − e2 − 6e6 + 12e7 }, C0 = Ae1 + Ad e2 , C12 = col{e0 , e0 , e6 }, C22 = col{e0 , e0 , −e4 }, C41 = col{e8 , e2 , e1 , e2 , e3 , e0 }, C50 = col{e1 − e2 , e0 , e0 , e0 , e0 , e0 }, C52 = col{e0 , e0 , e0 , e0 , e0 , e6 − e7 }, C71 = col{e9 , e3 , e1 , e2 , e3 , hM e4 }, C80 = col{e2 − e3 , e0 , e0 , e0 , e0 , e0 } C82 = col{e0 , e0 , e0 , e0 , e0 , e4 − e5 },

C11 = col{e1 , e2 , e0 }, C21 = col{e2 , e3 , hM e4 }, C3 = col{C0 , e1 , e1 , e2 , e3 , e0 }, C42 = col{e0 , e0 , e0 , e0 , e0 , e6 }, C51 = col{e0 , e6 , e1 , e2 , e3 , e0 }, C6 = col{e8 , e2 , e1 , e2 , e3 , e0 }, C72 = col{e0 , e0 , e0 , e0 , e0 , −e4 }, C81 = col{e0 , e4 , e1 , e2 , e3 , e0 },

D1 = col{C0 , (1 − h˙ (t ))e8 , e1 − (1 − h˙ (t ))e2 }, D2 = col{(1 − h˙ (t ))e8 , e9 , (1 − h˙ (t ))e2 − e3 }, D3 = col{e0 , e0 , C0 , (1 − h˙ (t ))e8 , e9 , e1 }, D4 = col{e0 , e0 , C0 , (1 − h˙ (t ))e8 , e9 , (1 − h˙ (t ))e2 }, and ei = [0n×(i−1 )n In 0n×(9−i )n ], for i = 1, 2, . . . , 9, and e0 = 0n×9n . Proof. The Lyapunov-Krasovskii functional condition V (xt , x˙ t ) > 0 is guaranteed by imposing Si (t )  0, Qi 0, and Ri 0 for i = 1, 2. Note that Si (t ) is an affine function in h(t) ∈ [0, hM ], then Si (t )  0 is ensured by imposing

Si2  0, hM Si1 + Si2  0, for i = 1, 2. Hereafter, we show that the Lyapunov-Krasovskii derivative condition V˙ (xt , x˙ t ) < 0 is satisfied if the inequalities in (9) and (10) hold. For a compact notation we define the augmented vector

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 )}, with

ρ1 (t ) =



t −h(t )

t−hM



x (s ) ds, hM − h(t )

x (s ) ρ3 (t ) = ds, and t −h(t ) h (t ) t

ρ2 (t ) = ρ4 (t ) =





t −h(t )

t−hM t

t −h(t )

(t − h(t ) − s )x(s ) ds, (hM − h(t ))2 (t − s )x(s ) ds. h2 (t )

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Taking the time derivative of (7) yields

V˙ (xt , x˙ t ) = 2η˙ 1T (t )S1 (t )η1 (t ) + η1T (t )S˙ 1 (t )η1 (t ) + 2η˙ 2T (t )S2 (t )η2 (t ) + η2T (t )S˙ 2 (t )η2 (t ) − (1 − h˙ (t ))η3T (t , t − h(t ))Q1 η3 (t , t − h(t ))  t ∂ + η3T (t , t )Q1 η3 (t , t ) + 2 η3T (t, s )dsQ1 η3 (t, s ) + (1 − h˙ (t ))η4T (t , t − h(t ))Q2 η4 (t , t − h(t )) ∂ t t −h(t )  t −h(t ) ∂ − η4T (t, t − hM )Q2 η4 (t, t − hM ) + 2 η4T (t, s )dsQ2 η4 (t, s ) ∂t t−hM + h2M x˙ T (t )R2 x˙ (t ) + (1 − h˙ (t ))(h2M − h(t )hM )x˙ T (t − h(t ))(R1 − R2 )x˙ (t − h(t ))

 t −h(t )   t − hM x˙ T (s )R1 x˙ (s )ds + x˙ T (s )R2 x˙ (s )ds , t −h(t )

t−hM

(14)

where η˙ 1 (t ) = ξ T (t )D1 , η˙ 2 (t ) = ξ T (t )D2 , η3 (t , t ) = C3 ξ (t ), η3 (t , t − h(t )) = (C41 + h(t )C42 )ξ (t ), η4 (t , t − h(t )) = C6 ξ (t ), η4 (t, t − hM ) = (C71 + h(t )C72 )ξ (t ), and



∂ η3 (t, s ) = ξ T (t )(C50 + h(t )C51 + h2 (t )C52 )Q1 D3 ξ (t ), ∂ t t −h(t )  t −h(t ) ∂ η4T (t, s )dsQ2 η4 (t, s ) = ξ T (t )(C80 + α hM C81 + (hM − h(t ))2 C82 )Q2 D4 ξ (t ), ∂t t−hM t

η3T (t, s )dsQ1

with α = (hM − h(t ))/hM . Under the assumption that R1 0 and R2 0, an upper bound for the last two integral terms in (14) can be obtained by applying Lemma 3, which leads to



t −h(t )

hM t−hM

x˙ T (s )R1 x˙ (s )ds +



t

t −h(t )



x˙ T (s )R2 x˙ (s )ds



1

α

ξ T (t )1T R˜1 1 ξ (t ) +

1 ξ T (t )2T R˜2 2 ξ (t ), 1−α

where R˜i = diag{Ri , 3Ri , 5Ri } for i = 1, 2 and  1 ,  2 are given in Theorem 1. Hence, we can apply Lemma 4 to obtain

1

α

ξ T (t )1T R˜1 1 ξ (t ) +

1 ξ T (t )2T R˜2 2 ξ (t )  ξ T (t ){1T [R˜1 + (1 − α )X1 ]1 + 21T [αY1 + (1 − α )Y2 ]2 1−α

+ 2T (R˜2 + α X2 )2 }ξ (t ).

Thus based on the previous two inequalities we have that

V˙ (xt , x˙ t ) ≤ ξ T (t )[h2 (t ) 2 (h˙ (t )) + h(t ) 1 (h˙ (t )) + 0 (h˙ (t ))]ξ (t ), with 0 (h˙ (t )), 1 (h˙ (t )), and 2 (h˙ (t )) given in (11), (12), and (13), respectively. Therefore, under the constraint (9), the Lyapunov-Krasovskii derivative condition V˙ (xt , x˙ t ) < 0 is satisfied if (10) holds, concluding this proof.  The previous theorem constitutes the main result in this paper, however evaluate the condition in (10) within the finite interval for all h(t) ∈ [0, hM ] is not practical. In order to eliminate the dependence on the variable h(t), we invoke Lemmas 1 and 6 to translate the Theorem 1 into a finite dimensional convex optimization problem. We emphasize that the Lemma 6 leads to an exact LMI characterization of the quadratic condition in (10). Accordingly the following two corollaries are provided. The first one is the combination of Theorem 1 with Lemma 1, setting a2 = ξ T (t )ϒ2 (h˙ (t ))ξ (t ), a1 = ξ T (t )ϒ1 (h˙ (t ))ξ (t ), and a0 = ξ T (t )ϒ0 (h˙ (t ))ξ (t ). Corollary 1. For given scalars d1 , d2 , and hM , system (1) is asymptotically stable if there exist matrices S11 ∈ S3n , S12 ∈ S3+n , S21 ∈ S3n , S22 ∈ S3+n , Qi ∈ S6+n , Ri ∈ Sn+ , Xi ∈ S3n , Yi ∈ R3n×3n , for i = 1, 2 such that (8), (9), and the following LMI conditions hold for j = 1, 2.

0 ( d j ) ≺ 0 ,

h2M 2 (d j ) + hM 1 (d j ) + 0 (d j ) ≺ 0, −h2M 2 (d j ) + 0 (d j ) ≺ 0

where 0 (h˙ (t )), 1 (h˙ (t )), and 2 (h˙ (t )) are given in Theorem 1. The next corollary results from the combination of Theorem 1 with Lemma 6. Corollary 2. For given scalars d1 , d2 , and hM , system (1) is asymptotically stable if there exist matrices S11 ∈ S3n , S12 ∈ S3+n , S21 ∈ S3n , S22 ∈ S3+n , Qi ∈ S6+n , Ri ∈ Sn+ , Di ∈ S9+n , Xi ∈ S3n , Yi ∈ R3n×3n and skew-symmetric matrices Gi ∈ R9n×9n for i = 1, 2, such that (8), (9), and the following conditions hold for j = 1, 2.



0 ( d j ) 1

(d j ) 2 1

  T 

1 ( d j ) C −D j − J ∗

2 ( d j )

1 2

Gj Dj

 

C ≺ 0, J

(15)

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Table 1 Maximum admissible upper bound hM of the delay h(t) for given d1 = −d2 and number of decision variables (NoV). Example 1. Method

d2

[3, IQC analysis] [25, Theorem 1] [44, Proposition 1] [49, Proposition 2] [39, Theorem 1] [50, Theorem 1] Corollary 1 Corollary 2





where C = (hM /2 )I9n respectively.



09n , J = (hM /2 )I9n

NoV

0.1

0.5

0.8

4.714 4.829 4.910 4.929 4.921 4.942 4.938 5.044

2.280 3.155 3.233 3.309 3.221 3.252 3.281 3.443

1.608 2.730 2.789 2.882 2.792 2.823 2.854 2.983

5.5n2 + 2.5n + 1 114n2 + 18n 54.5n2 + 6.5n 108n2 + 12n 115n2 + 10n 216n2 + 11n 82n2 + 16n 235n2 + 34n



−I9n , and 0 (h˙ (t )), 1 (h˙ (t )), and 2 (h˙ (t )) given in (11), (12), and (13),

The previous corollaries, as we discussed before, present numerical tractable LMI conditions to attest the quadratic condition in (10). As a result of Lemma 6 the Corollary 2 is less conservative, but it requires the determination of four new additional matrices Di and Gi , for i = 1, 2. In other words, the numerical complexity due to the application of Lemma 6 is significantly increased with the addition of 162n2 scalar decision variables. In short, Corollary 2 requires the computation of more than twice the number of scalar variables required by Corollary 1, see Table 1. Therefore, the improvement in Corollary 2 over Corollary 1 came with a considerable increase in the numerical complexity, especially for high-dimensional systems. 4. Numerical examples In this section, to illustrate the effectiveness of the proposed results, the stability of two benchmark systems drawn from the literature is studied. The computations are carried out in MATLAB using the LMI parser YALMIP [48] and the SDP solver Sedumi. The results are compared with those from some recent methods presented in the literature. Example 1. Consider the system in (1) with the following data:



−2 A= 0



0 , −0.9



−1 Ad = −1



0 . −1

The test performed is, for given bounds for the delay derivative h˙ (t ) ∈ [d1 , d2 ] setting d1 = −d2 , we search for the maximum admissible delay upper bound hM such that the LMI conditions in Corollaries 1 or 2 are feasible. Similarly for comparison purpose we apply the methods presented in [3,25,44,45,49]. The results are summarized in Table 1. Moreover, the number of decision variables (NoV) of each method is also listed in Table 1. The numerical results presented in Table 1 show that the proposed conditions presented in Corollary 1 outperforms most of recent results from the literature, which reveals the conservatism reduction merit of the proposed functional. The method in [50] can achieves larger upper bounds than Corollary 1, but it is important to mention that the conditions in [50] are obtained using a third-order Bessel-Legendre inequality. On the other hand, the results of Corollary 2 are less conservative than all of the methods listed in Table 1, even those based on the Bessel-Legendre inequality of higher orders [49,50]. Note that the improvement in Corollary 2 over Corollary 1 is due to Lemma 6. Comparing the number of decision variables (NoV) of each method in Table 1 we find that the achieved improvement in the proposed conditions came at a cost in computational complexity. It is also worth noting that Corollary 1 needs less decision variables than the methods in [25,49,50]. Remark 1. The number of decision variables of Corollary 1 can be reduced to 73n2 + 13n by choosing X1 = R1 − Y1 R−1 YT 2 2 −1 T and X2 = R2 − Y2 R1 Y1 in Lemma 4 and applying Schur complement. In [44] it is demonstrated that this choice for X1 e X2 is nonconservative. The same procedure, however, can not be applied to reduce the number of variables of Corollary 2, because in this case it is not possible to apply Schur complement due to the structure of the resulting stability condition. Example 2. In this example consider the matrices A and Ad in (1) given by:



0 A= −1



1 , −2



0 Ad = −1



0 . 1

For this system, performing the same tests in the previous example we obtain the results listed in Table 2, which show that among the methods presented in the Table, the proposed criteria in Corollary 2 can lead to the less conservative results. Please cite this article as: F.S.S. de Oliveira and F.O. Souza, Further refinements in stability conditions for time-varying delay systems, Applied Mathematics and Computation, https://doi.org/10.1016/j.amc.2019.124866

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F.S.S. de Oliveira and F.O. Souza / Applied Mathematics and Computation xxx (xxxx) xxx Table 2 Maximum admissible upper bound hM of the delay h(t) for given d1 = −d2 . Example 2. Method

[3, IQC analysis] [45, Theorem 1] [25, Theorem 1] [44, Proposition 1] [39, Theorem 1] [50, Theorem 1] Corollary 1 Corollary 2

d2 0.1

0.2

0.5

0.8

6.494 6.668 7.176 7.230 7.308 7.400 7.307 7.685

3.479 3.753 4.543 4.556 4.670 4.795 4.655 4.969

0.886 1.542 2.496 2.509 2.664 2.717 2.612 2.774

0.439 1.263 1.922 1.940 2.072 2.089 2.023 2.117

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