New versions of Bessel–Legendre inequality and their applications to systems with time-varying delay

Applied Mathematics and Computation 375 (2020) 125060

Contents lists available at ScienceDirect

Applied Mathematics and Computation journal homepage: www.elsevier.com/locate/amc

New versions of Bessel–Legendre inequality and their applications to systems with time-varying delay Jun Chen a,b, Ju H. Park b,∗ a b

School of Electrical Engineering and Automation, Jiangsu Normal University, Xuzhou 221116, China Department of Electrical Engineering, Yeungnam University, 280 Daehak-Ro, Gyongsan 38541, Republic of Korea

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 21 August 2019 Revised 9 December 2019 Accepted 12 January 2020

This paper is concerned with the stability of linear time-delay systems by applying the Lyapunov–Krasovskii (L–K) functional method. Two versions of Bessel–Legendre (B-L) inequality are developed that are more suitable to deal with the stability problem of systems with a time-varying delay. Meanwhile, in order to take full advantage of the interest of the new versions of B-L inequality, a novel L–K functional is properly tailored by integrating the integral information of the state into quadratic and integral terms. As a result, more relaxed stability conditions are obtained, whose effectiveness is illustrated by two commonly-used numerical examples.

Keywords: Bessel–Legendre inequality Lyapunov–Krasovskii functional Stability Time-delay system

© 2020 Elsevier Inc. All rights reserved.

1. Introduction Time delays widely exist in many fields of science and engineering. Their existence usually does harm to the stability of real-world systems [1–4]. Therefore, the last decades have witnessed a tremendous amount of research in the stability analysis of time-delay systems. Let us consider the following linear system with a time-varying delay described by:



x˙ (t ) = Ax(t ) + Ad x(t − h(t )), x(t ) = φ (t ), t ∈ [−hM , 0]

t≥0

(1)

where x(t ) ∈ Rn is the state vector; A and Ad are constant system matrices; φ (t) is the initial condition; h(t) is the timevarying delay that satisfies:

0 ≤ h(t ) ≤ hM ,

dm ≤ h˙ (t ) ≤ dM < 1.

(2)

The stability problem of system (1) is to achieve less conservative stability criteria by applying the Lyapunov–Krasovskii (L– K) functional method so that system (1) is guaranteed stable with h(t) satisfying the constraints (2), where the maximum allowable upper bound (MAUB) hM is expected to be as large as possible with given dm and dM . To handle integral terms arising in the time derivative of L–K functionals, the L–K functional method is usually combined with other techniques so that LMI-based stability conditions could be achieved. Until now, many techniques have been reported in the literature [5– 29], such as Moon’s inequality [5], delay decomposition [6,7], free-weighting matrices [8,9], the Jensen inequality [2], the Wirtinger-based inequality [10], free-matrix-based inequalities [11–15] and other inequalities [16–26]. ∗

Corresponding author. E-mail addresses: [email protected] (J. Chen), [email protected] (J.H. Park).

https://doi.org/10.1016/j.amc.2020.125060 0 096-30 03/© 2020 Elsevier Inc. All rights reserved.

2

J. Chen and J.H. Park / Applied Mathematics and Computation 375 (2020) 125060

Owing to its straightforwardness and conciseness, the bounding inequality technique has been attracting growing attention in recent years. As a powerful tool, the Jensen inequality is widely used to tackle the stability problem of time-delay systems [2]. However, due to its inherent conservatism [30], the Wirtinger-based integral inequality was proposed that encompasses the Jensen one [10]. Based on the Wirtinger-based inequality, a more relaxed stability condition was consequently obtained by combining with an appropriately constructed L–K functional. Later based on Legendre polynomials, a more general integral inequality, called Bessel–Legendre (B-L) inequality, was developed [31]. With more Legendre polynomials considered, more accurate bounds can be produced. However, although the B-L inequality is successfully adopted to the stability analysis of systems with a constant delay, there remain some difficulties in applying the B-L inequality. This is because the Legendre polynomials are involved in the B-L inequality. Moreover, the Legendre polynomials are very complex that are not easy to deal with. Therefore, how to transform the B-L inequality to an easily-utilized version is the first motivation of this paper. It is worth pointing out that another series of integral inequalities was proposed in [32] by constructing orthogonal polynomials. For the same reason, with polynomials involved, this kind of inequality is also not convenient to use. Apart from the use of a more accurate integral inequality, to construct a proper L–K functional is equally important [33,34]. Less conservatism of stability conditions may not be achieved unless the L–K functional is appropriately tailored [35–38]. When the Wirtinger-based inequality was utilized, the single-integral information of the state x(t) over the delay interval was integrated into the L–K functional [10]. As a result, the obtained stability condition shows improvement over those based on the Jensen one. In the same way, the double-integral information of the state x(t) over the delay interval was integrated into the L–K functionals [32,35] so as to coordinate with the second-order B-L inequality. However, it is observed that the attention of coordination was mainly focused on the quadratic terms in L–K functionals. In this case, the interest of the first- and second-order B-L inequalities was not fully considered. Therefore, how to construct a proper L–K functional so that the interest of the B-L inequality is fully considered is the second motivation. This paper aims at applying B-L inequality to the stability analysis of systems with a time-varying delay. The main contributions are listed as follows: (i) two versions of B-L inequality are newly developed that are more suitable to address the stability problem of systems with a time-varying delay; (ii) To take full advantage of the interest of the new versions of B-L inequality, a new L–K functional is properly tailored, in which more information of the state is integrated into both quadratic and integral terms; (iii) By employing the new versions of B-L inequality and novel L–K functionals, LMI-based conditions are achieved which are more relaxed than previous ones. To end this section, a quadratic lemma and an improved reciprocally convex lemma are recalled. Lemma 1 [32]. For a given quadratic function f (s ) = a2 s2 + a1 s + a0 , where a2 , a1 , a0 ∈ R, if f(0) < 0, f(h) < 0 and −a2 h2 + f (0 ) < 0, then the inequality f(s) < 0 holds for any s ∈ [0, h]. Lemma 2 [35]. For scalars α , β ∈ (0, 1) satisfying α + β = 1, matrices R1 , R2 ∈ Sn+ and Y1 , Y2 ∈ Rn×n , the inequality

1

α R1

∗





0 R1 + β X1 ≥ ∗ β R2 1

Y R2 + α X2



holds, where X1 = R1 − Y1 R−1 Y T and X2 = R2 − Y2T R−1 Y , Y = αY1 + β Y2 . 2 1 1 2 Notations. Throughout this paper, the notations are conventional. For example, Sn+ means the set of n × n matrices that

are symmetric and positive definite. N and N+ stand for non-negative and positive integers, respectively.

k l

refers to the

binomial coefficient given by (k−lk!)!l ! . 2. B-L inequality and its versions In [31], Legendre polynomials Lk (s) of degree k are defined as follows:

Lk (s ) :=

k 

lk

s

l=0

a

ba

l

,

s ∈ [a, b]

(3)

where sa := s − a, ba := b − a and lk := (−1 )k+l

kk+l l

l

. It is noted that this kind of Legendre polynomials is defined about

the monomial ( bsa )l . In fact, there is another kind of Legendre polynomials which is defined about the monomial ( bbs )l : a

Lk (s ) :=

k  l=0



ρlk

bs ba

a



l

,

s ∈ [a, b]

 

(4)



where bs := b − s and ρlk := (−1 )l kl k+l l . It is pointed out that the coefficient (−1 )k in lk can be removed, which does not make a difference to consequent results. In other words, the rest of polynomials are still orthogonal to each other. Lemma 3 [31]. For scalars N ∈ N, a < b ∈ R, a matrix R ∈ Sn+ and a function x(t ) : [a, b] → Rn , the inequality

a

b

xT (s )Rx(s )ds ≥

N  2k + 1 T k R k ba k=0

(5)

J. Chen and J.H. Park / Applied Mathematics and Computation 375 (2020) 125060

b

holds, where k =

a

3

Lk (s )x(s )ds.

Remark 1. Based on Legendre polynomials Lk (s), B-L inequality (5) was obtained. In fact, the B-L inequality still holds with

b Lk (s) being replaced by Lk (s ). In the sequel if not specified, k is referred to the definition k := a Lk (s )x(s )ds. It is clear that more Legendre polynomials considered, more accurate bounds produced. However, due to the complexity of Legendre polynomials, it is difficult to directly apply (5) to deal with the stability problem of time-delay systems. So in what follows, we will present two other versions of B-L inequality: one is without Legendre polynomials involved; the other is only with monomials involved. Both versions are easier to use than the original version (5). Before proceeding, we define:



0m(a,b) :=

b s1

a



b

a s1

Sm := a ] g[(ia,b ) :=



b

b



···



a



˙ i :=

a



···

bs ba

sm−1

a

sm−1

d sm · · · d s2 d s1 ,

a i



x(s )ds,

Li (s )x˙ (s )ds,

a

x ( sm )d sm · · · d s2 d s1 ,

xb := x(b),

xa := x(a ),

where m ∈ N+ , i ∈ N. The subscript (a, b) may be removed when no confusion is possible. Lemma 4. For scalars a, b ∈ R, i ∈ N, m ∈ N+ , a function x(t ) : [a, b] → Rn , the equations hold: ( ba )i [i] g

0i+1 =

where

(6)

i!

m Sm = (bma )!

(7)

˙ i = i ϑi

(8)

 ⎧ I −I , ⎪ ⎨ i =  i  ⎪ ⎩ ρ0i I − ρli I

i = 0,

 ρ1i I

l=0

ρ2i I

col {x , x }, ϑi = col {ϑb , a1  , . . . , 1  }, 0 S1 01 0i Si

···

ρii I ,

(9) i ≥ 1,

i = 0, i ≥ 1.

(10)

Proof of Eq. (6). Mathematical induction is used. With i = 0, Eq. (6) obviously holds. With i = k − 1 ≥ 1, we assume (6) holds. Then from the assumption, we have with i = k:

0k+1 = = where x˜(s ) =



a

a



···

(ba )k−1 ( k − 1 )!

s a

0k+1 =

b s1

sk

b

x(sk+1 )dsk+1 · · · ds2 ds1

a

a



bs ba

k−1

x˜(s )ds

x(v )dv. It follows from integration by parts that:

−1 ( k − 1 )!



b a

x˜(s )d

( bs )k k

=

( ba )k k!

g[k] . 

Proof of Eq. (7). Replacing x(s) by 1 in Eq. (6) with i = m − 1, we have:

(ba )m−1 Sm = ( m − 1 )!



b a



bs ba



m−1

ds =

( ba )m m!

. 

Proof of Eq. (8). With the fact that L0 (s ) = 1, we have:

˙ 0 =



b a

L0 (s )x˙ (s )ds = 0 ϑ0 .

4

J. Chen and J.H. Park / Applied Mathematics and Computation 375 (2020) 125060

With i ≥ 1, we have:



˙ i =

i b

a

ρ0i

=



ρli

l=0



b a

bs ba

l

x˙ (s )ds +

x˙ (s )ds

i 

ρli



i 

b



a

l=1

ρ0i (xb − xa ) +

=



bs ba

ρ0i xb −

i 

ρli xa +

i 

ρli

0 l

l=1

l [l−1] g ba

=

x˙ (s )ds

ba

l=0

Note that the fact

l

 l ρli −xa + g[l−1]

l=1

=



0 l Sl

= i ϑi .

Sl

has been considered.



˙ i is expressed as a linear combination of x(b), x(a) and  l /Sl , Remark 2. Eq. (8) is an important formula, in which  0 l ∈ {0, 1, . . . , i}. Especially, for any value of i, this expression can be directly calculated via (9). For example, with i = 3, it follows that ρ00 = 1, ρ01 = 1, ρ11 = −2, ρ02 = 1, ρ12 = −6, ρ22 = 6, ρ03 = 1, ρ13 = −12, ρ23 = 30 and ρ33 = −20. Then, the first four Legendre polynomials are listed: L0 (s ) = 1, L1 (s ) = 1 − 2κ (s ), L2 (s ) = 1 − 6κ (s ) + 6κ (s )2 and L3 (s ) = 1 − 12κ (s ) + 30κ (s )2 − 20κ (s )3 , where κ (s ) = bs /ba . Thus it follows from (9) that

 0 = I  2 = I



−I , −I

 1 = I I 

−6I

6I ,



−2I ,

 3 = I

−12I

I

30I



−20I .

From the above results, an interesting phenomenon is observed that all sums of each elements of i , i ∈ {0, 1, 2, 3}, are zero matrices. The underlying reason just can be explained by the formula (9). Based on Eq. (8), we get the first version of B-L inequality in which Legendre polynomials are completely removed. Lemma 5. For scalars N ∈ N, a < b ∈ R, a matrix R ∈ Sn+ and a function x(t ) : [a, b] → Rn , the inequality



b

a

x˙ T (s )Rx˙ (s )ds ≥

N

i=0

2i+1 ba

ϑiT Ti R i ϑi =

1 ba

˜ T R˜N ˜ N ϑN ϑNT N

(11)

holds, where i and ϑi are, respectively, defined in (9) and (10),

R˜N = diag{R, 3R, . . . , (2N + 1 )R},

   N = col N , N , . . . , N 0 1 N  N N

with N := N and i := i

0

···



0 of appropriate dimensions, 0 ≤ i < N.

b−s m−1 Eq. (6) shows that the multiple-integral term 0m and the single-integral term g[m−1] with the monomial ( b−a ) involved can be transformed into each other. Now, based on the formula (6) and Lemma 5, the second version of the B-L inequality is obtained.

Lemma 6. For scalars N ∈ N, a < b ∈ R, a matrix R ∈ Sn+ and a function x(t ) : [a, b] → Rn , the inequality

a

b

x˙ T (s )Rx˙ (s )ds ≥

N  2i + 1 T T T θ R i i θi b−a i i i i=0

=

1 ˜ T R˜N ˜ N N θN θ T T b−a N N N

holds, where

i =



diag{I, I}, diag{ 0 , I, 2I, . . . , iI},

i = 0, i ∈ N+

col {x , x }, θi = col {θb , 1a g[0] , 1 g[1] , . . . , 1 g[i] }, 0 ba ba ba Proof. According to the fact

l [l−1] g ba

=

0 l Sl

i = 0, i ∈ N+ .

, it follows with i ≥ 1 that

(12)

J. Chen and J.H. Park / Applied Mathematics and Computation 375 (2020) 125060

5

  1 1 1 ϑi = col ϑ0 , 01 , 02 , . . . , 0i S1 S2 Si   1 [0] 2 [1] i [i−1] = col ϑ0 , g , g , . . . , g = i θi . ba

ba

ba

We directly obtain (12) by substituting ϑi = i θi into (11).



Remark 3. From the two versions of B-L inequality, it is easily seen that Legendre polynomials are removed or substituted by simple polynomials. In the second version of B-L inequality, the denominators in the elements b1 g[i] (t ), i ∈ N, are the same. a

Meanwhile, the time derivative of g[i] (t) can be linearly represented by endpoints of x(t) or/and the terms g[0] (t ), . . . , g[i−1] (t ). All of these mentioned properties make the new versions of the B-L inequality more suitable and convenient to deal with the stability problem of time-delay systems.

3. Main results 3.1. A new L–K functional The following notations are defined for simplicity:

ht := h(t ),

hMt := hM − ht ,

  η (t ) := xt (0 ), xt (−ht ), xt (−hM ) ,

xt (0 ) := x(t ),

xt (−ht ) := x(t − ht ), xt (−hM ) := x(t − hM ), 0  −s i ] νi (t ) := g[(i−h ( t ) = xt (s )ds, ν¯ N (t ) := col {ν0 (t ), ν1 (t ), . . . , νN (t )}, t ,0 ) ht −ht

 −ht −ht − s i [i] μi (t ) := g(−hM ,−ht ) (t ) = xt (s )ds, μ ¯ N (t ) := col {μ0 (t ), μ1 (t ), . . . , μN (t )}, hMt −hM



ei,m := 0n×(i−1)n



0n×(m−i )n , i ∈ {1, . . . , m},

In×n



ξN (t ) := col η (t ), x˙ t (−ht ), x˙ t (−hM ),

ν0 (t ) μ0 (t ) ht

,

hMt

,...,

νN (t ) μN (t ) ht

,

hMt

 .

According to the above definitions, differentiating ν i (t) and μi (t) leads to: ν˙ i (t ) = −(1 − h˙ t )xt (−ht ) + i and μ˙ i (t ) = −xt (−hM ) + i(1 − h˙ t )

μ

μ˙ 0 (t ) = (1 − h˙ t )xt (−ht ) − xt (−hM ).

i−1 (t ) hMt





μ (t ) +ih˙ t hi



Mt

ν

i−1 (t ) ht



−ih˙ t

 ν (t ) i ht

with i ≥ 1. For i = 0, it follows that ν˙ 0 (t ) = xt (0 ) − (1 − h˙ t )xt (−ht ) and

Now, the new L–K functional is tailored with N ∈ N:

VN (t ) = V0,N (t ) + V1,N (t ) + V2,N (t )

(13)

where

V0,N (t ) = χNT (t )P χN (t ),

χN (t ) = col {η (t ), ν0 (t ), μ0 (t ), . . . , νN (t ), μN (t )}, V1,N (t ) =



t

η1T,N (t, s )Q1 η1,N (t, s )ds +

t−ht



t−ht

t−hM

η2T,N (t, s )Q2 η2,N (t, s )ds,

η1,N (t, s ) = col {x˙ (s ), x(s ), η (t ), ν¯ N (t )}, η2,N (t, s ) = col {x˙ (s ), x(s ), η (t ), μ¯ N (t )}, V2,N (t ) = hM



t

t−hM



s

t

x˙ T (u )Rx˙ (u )duds.

Remark 4. The novel functional VN (t) (13) is elaborately constructed. First, the construction of VN (t) is open. When the value of N rises, more integral information of the state x(t) is taken into account and more accurate integral inequalities could be utilized. Second, the integral information of the state is integrated into both the quadratic term V0,N (t) and the integral term V1,N (t). As a consequence, when the second version of the B-L inequality (12) is applied, the terms χ N (t), η1,N (t, s), η2,N (t, s) and their time derivatives can provide good cooperation. In this case, less conservative stability conditions should be expected.

6

J. Chen and J.H. Park / Applied Mathematics and Computation 375 (2020) 125060

3.2. Stability criteria (2N+5 )n Theorem 1. For given scalars dm , dM , hM ∈ R and N ∈ N, system (1) is asymptotically stable if there are matrices P ∈ S+ , (N+6 )n Q1 , Q2 ∈ S+ , R ∈ Sn+ and Y1 , Y2 ∈ R(N+2 )n×(N+2 )n such that

 ϒN ( 0 , d )

2T Y2T

(∗ )



< 0,

−R˜N

 ϒN ( h M , d ) (∗ )

1T Y1

(14)

 < 0,

−R˜N



(15)

−a2 (d )h2M + ϒN (0, d )

2T Y2T

(∗ )

−R˜N

 < 0,

(16)

hold for d ∈ {dm , dM }, where N = N + 1, ei = ei,7+2N ,

ϒN (ht , h˙ t ) = Sym{ T1,N (ht )P 2,N (h˙ t )} + 0 (ht , h˙ t ) + h2M eTs Res − 0,N (α ),

0,N (α ) = (2 − α )1T R˜N 1 + (1 + α )2T R˜N 2 + Sym{1T (αY1 + (1 − α )Y2 )2 }, T T T T a2 (h˙ t ) = c11 Q1 c11 − (1 − h˙ t )c21 Q1 c21 + (1 − h˙ t )c41 Q2 c41 − c51 Q2 c51 + Sym{D1T Q1 c32 + D2T Q2 c62 },

0 (ht , h˙ t ) = (c10 + ht c11 )T Q1 (c10 + ht c11 ) − (1 − h˙ t )(c20 + c21 ht )T Q1 (c20 + c21 ht ) + (1 − h˙ t )(c40 + hMt c41 )T Q2 (c40 + hMt c41 ) − (c50 + c51 hMt )T Q2 (c50 + c51 hMt ) + Sym{D1T Q1 (c30 + ht c31 + ht2 c32 )} + Sym{D2T Q2 (c60 + hMt c61 + h2Mt c62 )},

1,N (ht ) = col {eη , eν0 , eμ0 , . . . , eνN , eμN }, eη = col {e1 , e2 , e3 }, eνi = ht e6+2i , eν˙ i = eμ˙ i =



2,N (h˙ t ) = col {eη˙ , eν˙ 0 , eμ˙ 0 , . . . , eν˙ N , eμ˙ N },

eη˙ = col {es , (1 − h˙ t )e4 , e5 },

eμi = hMt e7+2i ,

i ∈ {0, 1, . . . , N },

e1 − (1 − h˙ t )e2 , i = 0, −(1 − h˙ t )e2 + ie6+2(i−1) − ih˙ t e6+2i ,

i ≥ 1,

(1 − h˙ t )e2 − e3 , i = 0 −e3 + i(1 − h˙ t )e7+2(i−1) + ih˙ t e7+2i ,

i ≥ 1,



es = Ae1 + Ad e2 ,

α = hMt /hM , 0N = col {0, . . . , 0}, 0η = col {0, 0, 0},   N+1

eν¯ N = col {e6 , e8 , . . . , e6+2N }, c10 = col {es , e1 , eη , 0N },

c20 = col {e4 , e2 , eη , 0N },

c21 = col {0, 0, 0η , eν¯ N },

c30 = col {e1 − e2 , 0, 0η , 0N }, c40 = col {e4 , e2 , eη , 0N }, c50 = col {e5 , e3 , eη , 0N },

eμ¯ N = col {e7 , e9 , . . . , e7+2N },

c11 = col {0, 0, 0η , eν¯ N },

c31 = col {0, e6 , eη , 0N },

c41 = col {0, 0, 0η , eμ¯ N }, c51 = col {0, 0, 0η , eμ¯ N },

c60 = col {e2 − e3 , 0, 0η , 0N },

c61 = col {0, e7 , eη , 0N },

D1 = col {0, 0, eη˙ , eν¯˙ N },

eν¯˙ N = col {eν˙ 0 , . . . , eν˙ N },

D2 = col {0, 0, eη˙ , eμ¯˙ N },

eμ¯˙ N = col {eμ˙ 0 , . . . , eμ˙ N },

T1N = col {e2 , e3 , e7 , e9 , . . . , e2(N −1)+7 }, ˜ N N T1N , 1 =

˜ N N T2N , 2 =

c32 = col {0, 0, 0η , eν¯ N },

c62 = col {0, 0, 0η , eμ¯ N },

T2N = col {e1 , e2 , e6 , e8 , . . . , e2(N −1)+6 }, R˜N = diag{R, 3R, . . . , (2N + 1 )R}.

Proof. Along the trajectory of system (1), differentiating Vi,N (t), i ∈ {0, 1, 2}, yields:

V˙ 0,N (t ) = ξNT (t )Sym{ T1,N (ht )P 2,N (h˙ t )}ξN (t ),

(17)

J. Chen and J.H. Park / Applied Mathematics and Computation 375 (2020) 125060

V˙ 1,N (t ) =

7

η1 (t , t )T Q1 η1 (t , t ) − (1 − h˙ t )η1 (t, t − ht )T Q1 η1 (t, t − ht ) t η (t, s ) +2 η1 (t, s )T Q1 1 ds dt

t−ht

+ (1 − h˙ t )η2 (t, t − ht )T Q2 η2 (t, t − ht ) − η2 (t, t − hM )T Q2 η2 (t, t − hM ) t−ht η (t, s ) +2 η2 (t, s )T Q2 2 ds dt t−hM =

ξN (t )T 0 (ht , h˙ t )ξN (t ),

(18)

V˙ 2,N (t ) = h2M x˙ (t )T Rx˙ (t ) − δ (t ),

(19)

where

χN (t ) = 1,N (ht )ξN (t ), χ˙ N (t ) = 2,N (h˙ t )ξN (t ), η1 (t , t ) = (c10 + c11 ht )ξN (t ), η1 (t , t − ht ) = (c20 + ht c21 )ξN (t ), t η (t, s ) 2 η1 (t, s )T Q1 1 ds = ξN (t )T Sym{D1T Q1 (c30 + c31 ht + c32 ht2 )}ξN (t ), dt

t−ht

η2 (t, t − ht ) = (c40 + c41 hMt )ξN (t ), η2 (t , t − hM ) = (c50 + hMt c51 )ξN (t ), t−ht η (t, s ) 2 η2 (t, s )T Q2 2 ds = ξN (t )T Sym{D2T Q2 (c60 + c61 hMt + c62 h2Mt )}ξN (t ), t−hM

δ (t ) = hM



dt

t

t−hM

x˙ (s )T Rx˙ (s )ds,

with terms such as 1,N (ht ), 2,N (h˙ t ) and cij are all defined in Theorem 1. Then, applying Lemmas 2 and 6 to δ (t) yields

δ (t ) = hM



−ht

−hM

x˙ t (s )T Rx˙ t (s )ds + hM

1



0 −ht

x˙ t (s )T Rx˙ t (s )ds

1

ξ (t )T 1T R˜N 1 ξN (t ) + ξN (t )T 2T R˜N 2 ξN (t ) α N 1−α ≥ ξN (t )T ( 0,N (α ) − 1,N (α ))ξN (t ) ≥

(20)

where α ,  1 , R˜N ,  2 and 0,N (α ) are defined in Theorem 1, and T T T ˜−1

1,N (α ) = (1 − α )1T Y1 R˜−1 N Y1 1 + α 2 Y2 RN Y2 2 .

From (17)–(20) it follows:

V˙ N (t ) = V˙ 0,N (t ) + V˙ 1,N (t ) + V˙ 2,N (t ) ≤

    ξNT (t ) YN ht , h˙ t + 1,N (α ) ξN (t )

where ϒN (ht , h˙ t ) is defined in Theorem 1. Similarly in [35], define the quadratic function f (ht ) := ζ T (ϒN (ht , h˙ t ) +

1,N (α ))ζ , ∀ζ ∈ R(2N+7)n . Then, one has

f (ht ) =

ζ T (ht2 a2 (h˙ t ) + ht a1 (h˙ t ) + a0 (h˙ t ))ζ

= a˜2 ht2 + a˜1 ht + a˜0 where a2 (h˙ t ) is defined in Theorem 1, a1 (h˙ t ) and a0 (h˙ t ) are symmetric matrices, irrespectively of ht and a˜i = ζ T ai (h˙ t )ζ , i ∈ {0, 1, 2}. According to Lemma 1 and the Schur complement, the inequality f(ht ) < 0 with ht satisfying (2) is guaranteed by (14), (15) and (16). So, by setting ζ = ξN (t ), one has V˙ N (t ) < 0 with ht satisfying (2). Finally, based on the L–K stability theorem, one gets a conclusion that system (1) is asymptotically stable. This completes the proof.  Remark 5. From the proof, it is found that the terms χ N (t), χ˙ N (t ), ν¯ N (t ), ν¯˙ N (t ), μ ¯ N (t ) and μ ¯˙ N (t ) in V˙ 0,N (t ) and V˙ 1,N (t ) are adequately represented by elements of ξ N (t), which makes it possible to take full advantage of the interest of the new versions of the B-L inequality. Remark 6. According to Lemma 6,  1 and  2 can be directly calculated. For instance, when N = 2, we have N = 3. Then, it follows that 1 = col {e2 − e3 , e2 + e3 − 2e7 , e2 − e3 − 6e7 + 12e9 , e2 + e3 − 12e7 + 60e9 − 60e11 } and 2 = col {e1 − e2 , e1 + e2 − 2e6 , e1 − e2 − 6e6 + 12e8 , e1 + e2 − 12e6 + 60e8 − 60e10 }, where ei = ei,11 . If the quadratic term V0,N (t) is removed from VN (t) (13), we have a special case of Theorem 1.

8

J. Chen and J.H. Park / Applied Mathematics and Computation 375 (2020) 125060 Table 1 The MAUBs hM for dM = −dm in Example 1. dM

0.1

0.5

0.8

K

[21] [32] [11] [35]

4.714 4.753 4.788 4.910

2.608 2.429 3.055 3.233

2.375 2.183 2.615 2.789

100 116 282 231

Theorem 2 (N = 0) Theorem 2 (N = 1) Theorem 2 (N = 2)

4.853 4.938 4.939

3.217 3.284 3.295

2.791 2.859 2.870

191 285 403

Theorem 1 (N = 0) Theorem 1 (N = 1) Theorem 1 (N = 2)

4.854 4.940 4.943

3.220 3.303 3.313

2.795 2.874 2.884

246 390 574

Table 2 The MAUBs hM for dM = −dm in Example 2. dM

0.1

0.2

0.5

0.8

[35] Theorem Theorem Theorem Theorem Theorem Theorem

7.230 7.214 7.326 7.362 7.215 7.333 7.398

4.556 4.584 4.685 4.767 4.584 4.696 4.818

2.509 2.521 2.642 2.707 2.531 2.668 2.738

1.940 1.940 2.051 2.081 1.959 2.067 2.103

2 2 2 1 1 1

(N (N (N (N (N (N

= = = = = =

0) 1) 2) 0) 1) 2)

Theorem 2. For given scalars hM , dm , dM ∈ R and N ∈ N, system (1) is asymptotically stable if there are matrices Q1 , Q2 ∈ (N+6 )n S+ , R ∈ Sn+ and Y1 , Y2 ∈ R(N+2 )n×(N+2 )n such that (14)–(16) hold with d ∈ {dm , dM }, where ϒN (ht , h˙ t ) = 0 (ht , h˙ t ) + h2M eTs Res − 0,N (α ), and other terms are all defined in Theorem 1.

s

s Remark 7. If the integral terms t−h and t−h are, respectively, included in η1,N (t, s) and η2,N (t, s), as done in [35], the t M conservatism of Theorems 1 and 2 is expected to further decrease. Remark 8. It is known that the computation complexity of LMI-based conditions can be estimated by the number L of LMI rows and the number K of decision variables. For instance, there are L = 72n and K = 54.5n2 + 6.5n in Proposition 1 [35], L = (18N + 54 )n and K = (3N 2 + 20N + 44.5 )n2 + (N + 6.5 )n in Theorem 2 and L = (18N + 54 )n and K = (5N 2 + 30N + 57 )n2 + (2N + 9 )n in Theorem 1. In the case of N = 1, Theorem 2 has the same number of the LMI rows with the other two conditions while a little more decision variables than Proposition 1 in [35] and much less decision variables than Theorem 1. Therefore, Theorem 2 may reach a good compromise between the conservatism and computation complexity, which will be clearly seen in the following numerical examples. Remark 9. It is noted that owing to the existence of diagonal terms such as −(1 − h˙ t )(c20 + c21 ht )T Q1 (c20 + c21 ht ), Theorems 1 and 2 are infeasible when dM ≥ 1. In this case, it is necessary to modify the L–K functional by removing some terms involving x(t − ht ). More details can be referred to [17]. 4. Numerical examples

Example 1. Consider system (1) with



A=

−2.0 0.0



0.0 , −0.9



Ad =

−1.0 −1.0



0.0 . −1.0

This example is checked by several stability conditions. The MAUBs and the numbers of decision variables are, respectively, shown in Table 1. It is found that MAUBs determined by Theorems 2 and 1 with N = 1 or N = 2 are all larger than those obtained by other conditions in [11,21,32,35]. Even with N = 0, the MAUBs determined by Theorem 2 are almost larger than those determined by other conditions. Especially, when dM = 0.8, the MAUB determined by Theorem 2 is larger than that in [35] while with a smaller number of decision variables. Meanwhile, it is not hard to found that, as the value of N increases to 2 from 1, the MAUBs obtained by Theorems 2 and 1 change a little while the numbers of decision variables increase sharply. As stated in Remark 8, Theorem 2 with N = 1 is more competitive, which is less conservative than previous conditions while with a relatively small number of decision conditions.

J. Chen and J.H. Park / Applied Mathematics and Computation 375 (2020) 125060

Example 2. Consider system (1) with



0.0 A= −1.0



1.0 , −2.0



0.0 Ad = −1.0

9



0.0 . 1.0

The comparison results among [35], Theorems 2 and 1 are listed in Table 2. As expected, the MAUBs obtained by Theorems 2 and 1 with N = 1 and N = 2 are larger than those obtained in [35]. Additionally, the MAUBs obtained by Theorem 1 are correspondingly larger than those obtained by Theorem 2 with different N, which shows that the quadratic function is helpful in reducing the conservatism. 5. Conclusions More relaxed stability conditions have been obtained for time-delay systems via new versions of the Bessel-Legendre inequality. A novel Lyapunov–Krasovskii functional has been deliberately constructed by integrating more information of the state into quadratic and single-integral terms. Finally, two numerical examples has been given to illustrate the effectiveness of the proposed approach. Acknowledgments This work of J. Chen was supported in part by the NSFC under Grant 61773186 and Grant 61877030, and in Part by the Science Fundamental Research Project of Jiangsu Normal University under Grant 17XLR045. Also, the work of J.H. Park was supported by Basic Science Research Programs through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (Grant number NRF-2017R1A2B2004671). References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15] [16] [17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29]

J.P. Richard, Time-delay systems, an overview of some recent advances and open Automatica, problems 39 (10) (2003) 1667–1694. K. Gu, V.L. Kharitonov, J. Chen, Stability of time-delay systems, Birkhäuser, Boston, USA, 2003. J.H. Park, T.H. Lee, Y. Liu, J. Chen, Dynamic Systems with Time Delays: Stability and Control, Springer-Nature, Singapore, 2019. X.M. Zhang, Q.L. Han, A. Seuret, F. Gouaisbaut, Y. He, Overview of recent advances in stability of linear systems with time-varying delays, IET Control Theory Appl. 13 (1) (2018) 1–16. Y.S. Moon, P. Park, W.H. Kwon, Y.S. Lee, Delay-dependent robust stabilization of uncertain state-delayed systems, Int. J. Control 74 (14) (2001) 1447–1455. Q.L. Han, A discrete delay decomposition approach to stability of linear retarded and neutral systems, Automatica 45 (2) (2009) 517–524. X. Meng, J. Lam, B. Du, H. Gao, A delay-partitioning approach to the stability analysis of discrete-time systems, Automatica 46 (3) (2010) 610–614. Y. He, Q.G. Wang, C. Lin, M. Wu, Delay-range-dependent stability for systems with time-varying delay, Automatica 43 (2) (2007) 371–376. S. Xu, J. Lam, Improved delay-dependent stability criteria for time-delay systems, IEEE Trans. Autom. Control 50 (3) (2005) 384–387. A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: application to time-delay systems, Automatica 49 (9) (2013) 2860–2866. H.B. Zeng, Y. He, M. Wu, J. She, Free-matrix-based integral inequality for stability analysis of systems with time-varying delay, IEEE Trans. Autom. Control 60 (10) (2015) 2768–2772. H.B. Zeng, X.G. Liu, W. Wang, A generalized free-matrix-based integral inequality for stability analysis of time-varying delay systems, Appl. Math. Comput. 354 (2019) 1–8. C.K. Zhang, Y. He, L. Jiang, W.J. Lin, M. Wu, Delay-dependent stability analysis of neural networks with time-varying delay: A generalized free-weighting-matrix approach, Appl. Math. Comput. 294 (2017) 102–120. H.B. Zeng, X.G. Liu, W. Wang, S.P. Xiao, New results on stability analysis of systems with time-varying delays using a generalized free-matrix-based inequality, J. Franklin Inst. 356 (13) (2019) 7312–7321. J. Chen, J.H. Park, S. Xu, Stability analysis for delayed neural networks with an improved general free-matrix-based integral inequality, IEEE Trans. Neural Netw. Learn. Syst. doi:10.1109/TNNLS.2019.2909350. H.B. Zeng, K.L. Teo, Y. He, W. Wang, Sampled-data-base dissipative control of TS fuzzy systems, Appl. Math. Modell. 65 (2019) 415–427. J. Chen, J.H. Park, S. Xu, Stability analysis of systems with time-varying delay: a quadratic-partitioning method, IET Control Theory Appl. 13 (18) (2019) 3184–3189. J. Chen, S. Xu, B. Zhang, Single/multiple integral inequalities with applications to stability analysis of time-delay systems, IEEE Trans. Autom. Control 62 (7) (2017) 3488–3493. J. Chen, S. Xu, B. Zhang, G. Liu, A note on relationship between two classes of integral inequalities, IEEE Trans. Autom. Control 62 (8) (2017) 4044–4049. J. Chen, J.H. Park, S. Xu, Stability analysis for neural networks with time-varying delay via improved techniques, IEEE Trans. Cybern. 49 (12) (2019) 4495–4500. C.K. Zhang, Y. He, L. Jiang, M. Wu, H. B. Zeng, Stability analysis of systems with time-varying delay via relaxed integral inequalities, Syst. Control Lett. 92 (2016) 52–61. E. Gyurkovics, G. Szabo-Varga, K. Kiss, Stability analysis of linear systems with interval time-varying delays utilizing multiple integral inequalities, Appl. Math. Comput. 311 (2017) 164–177. J. Lam, B. Zhang, Y. Chen, S. Xu, Reachable set estimation for discrete-time linear systems with time delays, Int. J. Robust Nonlin. Control 25 (2) (2015) 269–281. P.G. Park, J.W. Ko, C. Jeong, Reciprocally convex approach to stability of systems with time-varying delays, Automatica 47 (1) (2011) 235–238. M.J. Park, O.M. Kwon, J.H. Ryu, Generalized integral inequality: Application to time-delay systems, Appl. Math. Lett. 77 (2018) 6–12. M.J. Park, O.M. Kwon, S.G. Choi, Stability analysis of discrete-time switched systems with time-varying delays via a new summation inequality, Nonlinear Anal. Hybrid Syst 23 (2017) 76–90. X. Chen, T. Huang, J. Cao, J.H. Park, J. Qiu, Finite-time multi-switching sliding mode synchronisation for multiple uncertain complex chaotic systems with network transmission mode, IET Control Theory Appl. 13 (9) (2019) 1246–1257. H. Li, N. Zhao, X. Wang, X. Zhang, P. Shi, Necessary and sufficient conditions of exponential stability for delayed linear discretetime systems, IEEE Trans. Autom. Control 64 (2) (2019) 712–719. Z.M. Gao, Y. He, M. Wu, Improved stability criteria for the neural networks with time-varying delay via new augmented lyapunov-krasovskii functional, Appl. Math. Comput. 349 (2019) 258–269.

10

J. Chen and J.H. Park / Applied Mathematics and Computation 375 (2020) 125060

[30] C. Briat, Convergence and equivalence results for the jensen’s inequality—application to time-delay and sampled-data systems, IEEE Trans. Autom. Control 56 (7) (2011) 1660–1665. [31] A. Seuret, F. Gouaisbaut, Hierarchy of LMI conditions for the stability analysis of time-delay systems, Syst. Control Lett. 2 (81) (2015) 1–7. [32] J.H. Kim, Further improvement of jensen inequality and application to stability of time-delayed systems, Automatica 64 (2016) 121–125. [33] T.H. Lee, J.H. Park, S. Xu, Relaxed conditions for stability of time-varying delay systems, Automatica 75 (2017) 11–15. [34] T.H. Lee, J.H. Park, A novel lyapunov functional for stability of time-varying delay systems via matrix-refined-function, Automatica 80 (2017) 239–242. [35] X.M. Zhang, Q.L. Han, A. Seuret, F. Gouaisbaut, An improved reciprocally convex inequality and an augmented Lyapunov–Krasovskii functional for stability of linear systems with time-varying delay, Automatica 84 (2017) 221–226. [36] J.H. Kim, Note on stability of linear systems with time-varying delay, Automatica 47 (9) (2011) 2118–2121. [37] W. Kwon, B. Koo, S.M. Lee, Novel lyapunov–krasovskii functional with delay-dependent matrix for stability of time-varying delay systems, Appl. Math. Comput. 320 (2018) 149–157. [38] Z. Tang, J.H. Park, J. Feng, Impulsive effects on quasi-synchronization of neural networks with parameter mismatches and time-varying delay, IEEE Trans. Neural Netw. Learn. Syst. 29 (4) (2017) 908–919.