Multiple integral inequalities and stability analysis of time delay systems

Systems & Control Letters 96 (2016) 72–80

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Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle

Multiple integral inequalities and stability analysis of time delay systems É. Gyurkovics a,∗ , T. Takács b a

Mathematical Institute, Budapest University of Technology and Economics, Budapest, Pf. 91, 1521, Hungary

b

Corvinus University of Budapest, 8 Fővám tér, H-1093, Budapest, Hungary

highlights • New multiple integral inequalities are derived. • A set of sufficient LMI stability conditions for time delay systems are derived. • The LMI conditions are arranged into a bidirectional hierarchy.

article

info

Article history: Received 25 February 2016 Received in revised form 28 June 2016 Accepted 14 July 2016

Keywords: Integral inequalities Stability analysis Continuous-time delay systems Hierarchy of LMIs

abstract This paper is devoted to stability analysis of continuous-time delay systems based on a set of Lyapunov–Krasovskii functionals. New multiple integral inequalities are derived that involve the famous Jensen’s and Wirtinger’s inequalities, as well as the recently presented Bessel–Legendre inequalities of Seuret and Gouaisbaut (2015) and the Wirtinger-based multiple-integral inequalities of Park et al. (2015) and Lee et al. (2015). The present paper aims at showing that the proposed set of sufficient stability conditions can be arranged into a bidirectional hierarchy of LMIs establishing a rigorous theoretical basis for comparison of conservatism of the investigated methods. Numerical examples illustrate the efficiency of the method. © 2016 Elsevier B.V. All rights reserved.

1. Introduction Time delays are present in many physical, industrial and engineering systems. The delays may cause instability or poor performance of systems, therefore much attention has been devoted to obtain tractable stability criteria of systems with time delay during the past few decades (see e.g. the monographs [1–3], some recent papers [4–17] and the references therein). Several approaches have been elaborated and successfully applied for the stability analysis of time delay systems (see the references above for excellent overviews). Lyapunov method is one of the most fruitful fields in the stability analysis of time delay systems. On the one hand, more and more involved Lyapunov–Krasovskii functionals (LKF) have been introduced during the past decades. On the other hand, much effort has been devoted to derive more and more tight inequalities (Jensen’s inequality and different forms of Wirtinger’s inequality [1,2,4–14,18,19], etc.) for the estimation of quadratic single,

∗

Corresponding author. E-mail address: [email protected] (É. Gyurkovics).

http://dx.doi.org/10.1016/j.sysconle.2016.07.002 0167-6911/© 2016 Elsevier B.V. All rights reserved.

double and multiple integral terms in the derivative of the LKF. Simultaneously, augmented state vectors are introduced in part as a consequence of the improved estimations, in part on an ad hoc basis. The effectiveness of different methods is mainly compared using some numerical examples. Recently, the authors of [4,13] have introduced a very appealing idea of the hierarchy of LMI conditions offering a rigorous theoretical basis for comparison of stability LMI conditions. Based on Legendre polynomials, they proposed a generic set of single integral inequalities opening the way to the derivation of a set of stability conditions forming a hierarchy of LMIs. A further possibility for the derivation of improved stability conditions have been proposed by [5,6] using multiple integral quadratic terms in the LKF, together with Wirtinger-based multiple integral inequalities. Naturally the question arises: how these two lines of investigations are related to each other, and how sufficient stability conditions can be derived unifying the approaches of using multiple integral quadratic terms in the LKF and refined estimations of these integral terms. The aim of the present work is to answer these questions. On the one hand, multiple integral inequalities based on orthogonal hypergeometric polynomials will be derived that extend the results

É. Gyurkovics, T. Takács / Systems & Control Letters 96 (2016) 72–80

of [4,13] to multiple integrals and improve the estimations of [5,6]. On the other hand, a multi-parametric set of LMI conditions will be constructed, and it will be shown that a two parametric subset forms a bidirectional hierarchy of LMIs. Analogous results have been presented for discrete-time systems in [20]. The paper is organized as follows. In Section 2 it is shown, how the quadratic terms of the derivative of the LKF can be estimated by Bessel-type inequalities. It is also proven that these estimations relevantly improve a recently published result. A sufficient condition of asymptotic stability is presented in the form of an LMI in Section 3. The hierarchy of LMI conditions is established then in Section 4. Some benchmark numerical examples are shown in Section 5, the results of which are compared to earlier ones known from the literature. Finally, the conclusions will be drawn. The notations applied in the paper are very standard, therefore we mention only a few of them. Symbol A ⊗ B denotes the Kronecker-product of matrices A, B, while Sn and S+ n are the set of symmetric and positive definite symmetric matrices of size n × n, respectively. 2. Multiple integral inequalities 2.1. Preliminaries The paper deals with the stability analysis of the following continuous-time delay system x˙ (t ) = Ax(t ) + Ad1 x(t − τ ) + Ad2



(If ℓ = 0, then a single integral is considered.) Substitute s ∈ [a, b] by s = a + (b − a)x, where x ∈ [0, 1], and set Gi (x) = gi (a + (b − a)x), (i = 1, 2) on the right hand side of (5), then we obtain that

⟨g1 , g2 ⟩ℓ,[a,b] = (b − a)

x(s) ds,

t ≥ 0,

Thus it is sufficient to consider the orthogonal polynomials with respect to ⟨·, ·⟩ℓ,[0,1] . For any fixed non-negative integer ℓ, let us denote by Pℓ,n , (n = 0, 1, . . .) the polynomials of degree n orthogonal with respect to ⟨·, ·⟩ℓ,[0,1] . (For general theory see e.g. [21].) They can be given by the two parameters generalization of the Rodrigues-formula: Pℓ,0 (x) ≡ 1, Pℓ,n (x) =

(8)

1 1 d n!

xℓ

n



dxn

t ∈ [−τ , 0],

JW (f ) = ⟨f , Wf ⟩.

ℓ, n ≥ 0,

(2)

Lemma 1. If ν ≥ 0 is a given integer, then the following inequality holds 1

T  2 wj W wj , πj 

(4)

where wj = ⟨f , πj ⟩, and the scalar product is taken componentwise. Proof. The proof is standard, therefore it is omitted. (B.) Orthogonal hypergeometric polynomials. Suppose that ℓ ≥ 0 is a given integer and consider the closed interval [a, b]. For functions g1 , g2 ∈ L2 [a, b] define a scalar product by

⟨g1 , g2 ⟩ℓ,[a,b] =



s−a

ℓ

b−a

a

g1 (s)g2 (s) ds.

(5)

It is easy to see that ⟨g1 , g2 ⟩ℓ,[a,b] can equivalently be expressed as

⟨g1 , g2 ⟩ℓ,[a,b] =

ℓ! (b − a)ℓ if ℓ > 0.

b



b



b

··· a

v1

vℓ

(9)

(10) 1

 0

(iii) Pℓ,n (0) = (−1)n

2 xℓ Pℓ, n (x) dx =

ℓ+n n

g1 (s)g2 (s) ds dvℓ . . . dv1 ,

,

1

ℓ + 2n + 1

Pℓ,n (1) = 1.

pℓ,n (t ) = Pℓ,n



t −a

(11) (12)

 (13)

b−a

are orthogonal with respect to the scalar product (5), and b−a , ℓ + 2n + 1 ℓ+n pℓ,n (a) = (−1)n , pℓ,n (b) = 1.

 2 pℓ,n 

=

n

(14)

In what follows, polynomials qℓ,ℓ+j (x) = xℓ Pℓ,j (x) (for ℓ > 0, j = d 0, 1, . . .) and dx qℓ,ℓ+j (x) (for ℓ ≥ 0, j = 0, 1, . . .) have to be expressed in terms of the shifted Legendre polynomials P0,0 , P0,1 , . . . . To this end, consider the nonnegative integers n1 ≤ n2 , K , and introduce the notations Xn1 ,n2 = xn1 , xn1 +1 , . . . , xn2



T

,

T Πℓ,K (x) = Pℓ,0 (x), Pℓ,1 (x), . . . , Pℓ,K (x) , 

Dn1 ,n2 = diag {n1 , n1 + 1, . . . , n2 } , and G(ℓ, K ) ∈ R(K +1)×(K +1) with elements G(ℓ, K )1,1 G(ℓ, K )l,k = 0, if 1 ≤ l < k ≤ K + 1, k−1 l  l−j  ℓ+k+i , k − j i =1 i j=0

if l = 1, . . . K , k = 0, . . . , l, (6)

,

The polynomials

G(ℓ, K )l+1,k+1 = (−1)l+k



n = 1, 2, . . .

   n  ℓ+k+n k k n (i) Pℓ,n (x) = (−1) (−1) x , k ℓ+k

ℓ,[a,b]

b



n

 2 (ii) Pℓ,n ℓ,[0,1] =

(3)



xℓ (x2 − x)n ,

For ℓ = 0, this is the usual Rodrigues formula for the shifted Legendre polynomials. We note that polynomials (8)–(9) satisfy certain hypergeometrictype differential equation (see e.g. [22,23]). This is why they are frequently called ‘‘orthogonal hypergeometric polynomials’’. By straightforward calculation, it can be shown that they have the properties

(1)

(A.) A Bessel-type inequality. Let E be a Euclidean space with the scalar product ⟨·, ·⟩, and let πj ∈ E, (j = 0, 1, . . .) form an orthogonal system. Let n ≥ 1 be a given integer. For any f , g ∈ En , n + n define ⟨f , g ⟩ = i=1 ⟨fi , gi ⟩. Let W ∈ Sn . For any f ∈ E , consider the functional

j =0

(7)

k=0

t

where x(t ) ∈ R is the state, A, Ad1 and Ad2 are given constant matrices of appropriate size, the time delay τ is a known positive constant and x0 (·) is the initial function.

JW (f ) ≥

xℓ G1 (x)G2 (x) dx

= (b − a)⟨G1 , G2 ⟩ℓ,[0,1] .

nx

ν 

1

 0

t −τ

x0 (t ) = ϕ(t ),

73

(The void product equals to 1 by definition.)

= 1,

74

É. Gyurkovics, T. Takács / Systems & Control Letters 96 (2016) 72–80

Lemma 2. Let ℓ, K1 , K2 be given nonnegative integers satisfying the inequality K1 + ℓ ≤ K2 . Then xℓ Πℓ,K1 (x) = Ξℓ (K1 , K2 ) Π0,K2 (x)

(15)

Ξℓ (K1 , K2 ) = G(ℓ, K1 ) Oℓ,K1 ,K2 G(0, K2 ) ,   Oℓ,K1 ,K2 = 0K1 +1,ℓ IK1 +1 0K1 +1,K2 −(K1 +ℓ) , −1

(16)

where Wℓ = diag {(ℓ + 1), (ℓ + 3), . . . , (ℓ + 2νℓ + 1)} ⊗ W , b   T ΦMT = φ0T , . . . , φM p (s)f (s) ds, Ξℓ = −1 with φj = a 0,j Ξℓ (νℓ , M − 1) and Ξℓ (·, ·) is given by (16)–(17). Proof. Introduce the notation wℓ,j = ⟨f , pℓ,j ⟩ℓ,[a,b] (j 1, . . . , νℓ ) needed to apply Lemma 1. Clearly,

(17)

wℓ,j =

and

b

 a

 d  ℓ x Πℓ,K1 (x) = Zℓ (K1 , K2 ) Π0,K2 (x)

(18)

dx

ℓ,K1 ,K2 G(0, K2 )−1 , Zℓ (K1 , K2 ) = G(ℓ, K1 ) Dℓ,K1 +ℓ O  ℓ,K1 ,K2 = 0K1 +1,ℓ−1 O if ℓ > 0,



01,K1 IK1



qℓ,ℓ+j (x) =



0K1 +1,K2 −K1 +1 .

(21)

X0,K2 = G(0, K2 )

if ℓ ≥ 0, K1 ≥ 0,

Π0,K2 (x).

(22)

(23)

Since Xℓ,K1 +ℓ = Oℓ,K1 ,K2 X0,K2

(24)

for any K1 , K2 with K2 ≥ K1 + ℓ, we obtain relation (15)–(17) from (22)–(24). Furthermore, using the notation γ (x, ℓ) = xℓ−1 , if ℓ > 0, and γ (x, 0) = 0, we obtain that

 d  ℓ x Πℓ,K1 (x) = G(ℓ, K1 ) Dℓ,K1 +ℓ dx



γ (x, ℓ)

Xℓ,K1 +ℓ−1



Pℓ,j

s−a



b−a

f (s) ds.

(28)

,



ξjℓ,i P0,i (x),

(29)

  where ξ ℓ = ξjℓ,0 , . . . , ξjℓ,M −1 is the (j + 1)th row of matrix Ξℓ . j

Using the definition of φi and ΦM we obtain

wℓ,j =

and −1



i =0

M −1

Proof. It follows from (10) that xℓ Πℓ,K1 (x) = G(ℓ, K1 ) Xℓ,K1 +ℓ ,

ℓ

As we have seen, polynomials qℓ,ℓ+j (x) = xℓ Pℓ,j (x) appearing in (28) can be expressed as

(20)



b−a

M −1

0K1 +1,K2 +1−(K1 +ℓ) ,

and

0,K1 ,K2 = O

s−a

(19)



IK1 +1



= 0,



  ξjℓ,i φi = ξ ℓj ⊗ I ΦM .

(30)

i=0

Estimation (27) can be obtained from Lemma 1 by direct substitution taking into account (7) and (11).  Remark 1. We note that Lemma 3 is closely connected with Theorem 2.2 of [19]. Both results are based (explicitly or implicitly) on Lemma 1, thus they are substantially equivalent. The estimation of Lemma 3 may be more advantageous when it is applied for derivative of functions (see Lemma 4) and for stability analysis of time delay systems. The advantage is twofold: on the one hand, the variables are expressed using a common set of orthogonal polynomials independent of ℓ, on the other hand, the dependence on the length of the interval is relatively simple, since the matrices Ξℓ and Wℓ do not depend on b − a.

which yields (18)–(21) by taking into account that Xℓ−1,K1 +ℓ−1 =

ℓ,K1 ,K2 X0,K2 , if ℓ > 0, and 0 O 

case of ℓ = 0.

X0,K1 −1

T

0,K1 ,K2 X0,K2 in the = O

2.2. Integral inequalities

, [a, b] ⊂ R with b − a > 0 and 0 ≤ ℓ ∈ Z be given. Let W ∈ S+ n For any continuous f : [a, b] → Rn , consider the functional JW ,ℓ,a,b (f )

ℓ! = (b − a)ℓ

b



b

f (s)Wf (s) ds dvℓ . . . dv1 , T

··· v1

a

Gl (f , a, b, W ) =

b



vℓ

(25)

which can also be expressed as JW ,ℓ,a,b (f ) =

b





a

s−a

ℓ

b−a

(26)

Let νℓ ≥ 0 be a given integer. One can apply now Lemma 1 with E = L2[a,b] , the scalar product (5), ν = νℓ and πj = pℓ,j , (j = 0, 1, . . . , νℓ ). Now, our aim is to derive a lower estimation as a quadratic form with respect to variables independent of ℓ. Lemma 3. Let M > 0, ℓ ≥ 0 and νℓ ≥ 0 be given integers satisfying the condition ℓ + νℓ ≤ M − 1. Let JW ,ℓ,a,b (f ) be defined by (26). Then the following inequality holds true: 1 b−a

ΦMT (Ξℓ ⊗ I )T Wℓ (Ξℓ ⊗ I ) ΦM ,

(b − a)l JW ,l,a,b (f ). l!

(31)

To express the estimation of the present paper with the variables of [6], we need a short computation to show that wl,0 = (l+1)! l! g (f , a, b), and wl,1 = − (b−a)l Υl (f , a, b). Employing the (b−a)l l proposed estimation with (31), we obtain that

f T (s)Wf (s) ds

= ⟨f , Wf ⟩ℓ,[a,b] .

JW ,ℓ,a,b (f ) ≥

Remark 2. Paper [19] gives a thorough and detailed discussion of the relation between their WOPs-based result and the Jensen’s and Wirtinger’s inequalities published in a wide range of previous literature, therefore we only compare Lemma 3 to the recently published multiple integral inequality of Lemma 5 of [6]. Using the notations of [6], we can see that the relation of the investigated functionals can be given as

(27)

(l + 1)! T g (f , a, b)Wgl (f , a, b) (b − a)l+1 l l!(l + 3) + (l + 1)2 Υ T (f , a, b)W Υl (f , a, b). (b − a)l+1 l

Gl (f , a, b, W ) ≥

(32)

The first term of the lower bound of [6] is the same as in (32), while the second term in [6] is smaller than in (32), because the coefficient 1 appears in the estimation of [6] instead of (l + 1)2 of (32), thus the estimation of the present paper is tighter. The considerations above indicate that the results of [6] correspond to the choice of νl = 1, but the authors of [6] do not derive any estimation that can be characterized with νl > 1.

É. Gyurkovics, T. Takács / Systems & Control Letters 96 (2016) 72–80

Next, we shall derive a lower estimation also for the case, d when the functional is applied to the derivative f ′ (s) = ds f (s), i.e. consider JW ,ℓ,a,b (f ) = ′

b





s−a

ℓ

b−a

a

f (s) Wf (s) ds ′

′

T

= ⟨f ′ , Wf ′ ⟩ℓ,[a,b] .

(33)

Lemma 4. Let M ≥ 0, ℓ ≥ 0 and νℓ ≥ 0 be given integers satisfying the condition ℓ + νℓ ≤ max {0, M − 1}. Let JW ,ℓ,a,b (f ′ ) be defined by (33). Then the following inequality holds true: 1

JW ,ℓ,a,b (f ′ ) ≥

b−a

MT (Zℓ ⊗ I )T Wℓ (Zℓ ⊗ I ) Φ M , Φ

(34)

75

3. Stability analysis of continuous delayed systems Consider Eq. (1). Let M > 0, m1 ≥ 0, m2 ≥ 1 be given integers. Let xt (s) = x(t + s) be the solution of (1), and let φj (t ) and ΦM (t ) be defined for function f = xt as before with

φj (t ) =

0

{φ0 (t ), . . . , φM −1 (t )}. −τ p0,j (s)xt (s) ds, and ΦM (t ) = col Set furthermore  x(t ) = col {x(t ), ΦM (t )} ,   1 M (t ) = col x(t ), x(t − τ ), ΦM (t ) . Φ τ Consider the LKF candidate

M = col {f (b), f (a), where Wℓ is the same as in Lemma 3, Φ 1 1 0 = col {f (b), f (a)} , φ , . . . , φ , if M > 0 , Φ 0 M − 1 b −a b −a

V (xt , x˙ t ) = V1 (xt ) + V2 (xt ) + V3 (˙xt ),

Z0 = ϑ (ν10)

ϑ (ν20)

where

Zℓ = ϑ (ν1ℓ)

0νℓ





 −Z 0 ,  −Zℓ , if ℓ > 0, (1)

(35) V1 ( x t ) =  x(t )T P x(t ),

(2)

(1)

where the vectors ϑ k , ϑ k , 0k ∈ Rk+1 , are defined by ϑ k (2)

=

(1, . . . , 1) , 0k = (0, . . . , 0) , ϑ k = (−1, 1, . . . , ±1) and matrix Zℓ = Zℓ (νℓ , M − 1) is given by (19)–(21). T

T

T

Proof. Set θℓ,j = ⟨f ′ , pℓ,j ⟩ℓ,[a,b] , (j = 0, 1, . . . , νℓ , ) and apply Lemma 1. In order to obtain the estimation (34), we have to perform a short computation, as follows. Consider the previously introduced polynomials qℓ,ℓ+j again, then integrating by parts we obtain

θℓ,j =

 b

 a

s−a b−a

ℓ

pℓ,j (s)f ′ (s) ds

(36)

As we have seen, polynomials q′ℓ,ℓ+j having degree ℓ + j − 1 can be expressed by P0,0 , . . . , P0,ℓ+j−1 , P0,ℓ+j , . . . , P0,M −1 :



s−a b−a

M −1

 =



ζjℓ,i P0,i



i=0

s−a

M −1

 =

b−a



ζjℓ,i p0,i (s),

(37)

i =0

  where ζ ℓ = ζjℓ,0 , . . . , ζjℓ,M −1 is the (j + 1)th row of matrix Zℓ . Thus j θℓ,j = qℓ,ℓ+j (1)f (b) − qℓ,ℓ+j (0)f (a) −

1

M −1



b − a i =0

ζjℓ,i φi .

By straightforward calculation qℓ,ℓ+j (1) = 1, qℓ,ℓ+j (0) =

 (−1)j , 0,

V2 ( x t ) =

m1

0

j =0

−τ





P ∈ Snx (M +1) ,

s+τ

j

τ

(39)

xt (s)T Qj xt (s) ds,

Qj ∈ Snx , j = 0, . . . , m1 , +

V3 (˙xt ) = τ

m2  

0



s+τ

τ

−τ

j =1

(40)

j

x˙ t (s)T Rj x˙ t (s) ds,

Rj ∈ S+ nx , j = 1, . . . , m2 .

(41)

We note that V2 and V3 can also be written as multiple integrals (c.f. (25) and (26)).

= qℓ,ℓ+j (1)f (b) − qℓ,ℓ+j (0)f (0)    b s−a 1 ′ qℓ,ℓ+j f (s) ds. − b−a a b−a

q′ℓ,ℓ+j

(38)

Theorem 1. Let M > 0, m1 ≥ 0, m2 ≥ 1 and ν1,j ≥ 0, (j = 0, . . . , m1 ), ν2,k ≥ 0, (k = 0, . . . , m2 − 1) be given integers satisfying the inequalities j + ν1,j < M , k + ν2,k < M, for all j, k. System (1) is asymptotically stable, if there are matrices P ∈ + Snx (M +1) , Qj ∈ S+ nx , j = 0, . . . , m1 and Rk ∈ Snx , k = 1, . . . , m2 such that the LMIs

Ψm01 ,M (τ ) > 0,

(42)

ΨM1 (τ ) + Ψm21 ,M + Ψm3,21,M (τ ) − Ψm3,22,M (τ ) < 0 hold true, where m1 

Ψm01 ,M (τ ) = τ P +

 

diag 0, Ξj ⊗ I

T

(j)

Qj



Ξj ⊗ I



,

(43)

j =0

if if

ℓ = 0, ℓ > 0.

Estimation (34) can be obtained by direct substitution taking into account (7) and (11) and the definition of the variables.  Remark 3. Paper [19] gives the lower bound for functionals applied to derivatives of functions for several special cases together with comparisons with previously published estimations, therefore we refer the reader for discussions to [19]. We only mention that the relation between Lemma 4 and Lemma 6 of [6] is analogous to the one pointed out in Remark 2. Moreover neither [6] nor [19] derive any estimation for functionals applied to derivative of functions relying to polynomials of degree higher than 1.

ΨM1 (τ ) = ΓMT P ΛM + ΛTM P ΓM , m 1  2 Ψm1 ,M = diag Qj , −Q0 ,

(44)

j =0

−

m1  

j Ξj−1 ⊗ I

T

(j) Qj−1

 

Ξj−1 ⊗ I



,

(45)

j =1

Ψm3,21,M (τ ) = τ AT

m2 

Rj A,

(46)

j =1

Ψm3,22,M (τ ) =

m2  T (j)   1 j Zj−1 ⊗ I Rj−1 Zj−1 ⊗ I ,

τ

j =1

(47)

76

É. Gyurkovics, T. Takács / Systems & Control Letters 96 (2016) 72–80

matrices Ξk are given by Lemma 3 with νk = ν1,k , matrices Zk are given by Lemma 4 with νk = ν2,k , respectively, (k)

= diag {(j + 1), (j + 3), . . . , (j + (2M − 1))} ⊗ Qk ,   A = A, Ad1 , τ Ad2 , 0, . . . , 0 ∈ Rnx ×nx (M +2) ,     A 1 0 0 ΛM =  , ΓM = ⊗ I, 0 0 τ IM L0 ⊗ I  (1)  ) 0 = ϑ L ϑ (M2− −Z0 (M − 1, M − 1) , M −1 1   (j) Rj−1 = diag j, (j + 2), . . . , (j + 2νj−1 ) ⊗ Rj , Qj

(48)

The derivative of the first term of V2 is

dt

d

(52)

dt



) (2) ϑ (M1− 1 , ϑ M −1

0

(0) ≥ τ −1 x(t )T diag 0, (Ξ0 ⊗ I )T Q0 (Ξ0 ⊗ I )  x(t ).



(53)

If m1 > 0, apply (27) by taking νj = ν1,j ≤ M − j − 1 to the terms of V2 with j > 0. We obtain

−τ

s+τ

j

τ

s+τ

xt (s) Qj xt (s) ds T

τ

−τ

(54)

It follows from (53) and (54) that

 ≥ x(t )T P + τ −1

m1 



  T (j)    diag 0, Ξj ⊗ I Qj Ξj ⊗ I x(t ). (55)

Since V3 (˙xt ) ≥ 0, the existence of an appropriate µ1 follows from (55) and (42). d We shall prove next the negativity of dt V (xt , x˙ t ). Introduce the

notation V i (t ) = Vi (xt ), i = 1, 2, and V 3 (t ) = V3 (˙xt ), where xt is the solution of system (1). The derivative of the first term of (38) is

s−t +τ

j−1

τ

t −τ

x(s)T Qj x(s) ds

JQj ,j−1,−τ ,0 (xt ).

(59)

(60)

It follows from (60) that



d dt

0



s+τ

j

τ

−τ

xt (s)T Qj xt (s) ds ≤ x(t )T Qj x(t )

 T  1 1 − j ΦM (t )T Ξj−1 ⊗ I Qj(−j)1 Ξj−1 ⊗ I ΦM (t ), τ τ ˙ MT Ψm2 ,M Φ M . V 2 (t ) ≤ Φ 1

˙ V 3 (t ) = τ





φj (t ) =

 m2  d j=1

=τ

dt

m2  

p0,j (s)˙xt (s) ds = p0,j (0)x(t ) − p0,j (−τ )x(t − τ )

τ

s+τ

j =1

j

τ j

τ

x˙ t (s)T Rj x˙ t (s) ds =



JRj ,j−1,−τ ,0 (˙xt ) .

(62)

Applying now Lemma 4, it follows that JRj ,j−1,−τ ,0 (˙xt )

    M (t )T Zj−1 ⊗ I T Rj(−j)1 Zj−1 ⊗ I Φ M (t ), ≥ τ −1 Φ

(63)

 m  2  ˙V (t ) ≤ x˙ (t )T τ M (t )T Ψm3,2,M (τ )Φ M (t ). Rj x˙ (t ) − Φ 3 2

(64)

j =1

−τ

−

−τ



x˙ (t )T Rj x˙ (t ) −

0

1

0

where Rj−1 is given by (52). From (62) and (63) we obtain

d where  x˙ (t ) = col x˙ (t ), dt φ0 (t ), . . . , dtd φM −1 (t ) . The derivatives of φj s can be obtained by integration by parts:



(61)

(j)

˙

V 1 (t ) =  x˙ (t )T P x(t ) + xT (t )P x˙ (t ),



0

′

−τ

P0,j



s+τ

τ



xt (s) ds.

M −1

φj (t ) = x(t ) − (−1)j x(t − τ ) −

(56)

 l =0

1

ζj0,l φl (t ). τ

M (t ). On the other hand,  M (t ), Therefore,  x˙ (t ) = ΛM Φ x(t ) = ΓM Φ thus we obtain ˙ M (t )T ΨM1 (τ )Φ M (t ). V 1 (t ) = Φ

M (t ), (64) implies that Since x˙ (t ) = AΦ 

In consistence with (14), (37) and the definition of φl , it follows from (56) that

dt

j



JQj ,j−1,−τ ,0 (xt )

j =0

d

τ τ

t



j

Now compute the derivative of V 3 (t ). We obtain

V1 (xt ) + V2 (t )

dt

xt (s)T Qj xt (s) ds

which means that

  T (j)   ≥ τ −1 x(t )T diag 0, Ξj ⊗ I Qj Ξj ⊗ I  x(t ).

d

j

 T   ≥ τ −1 ΦM (t )T Ξj−1 ⊗ I Qj(−j)1 Ξj−1 ⊗ I ΦM (t ).





(58)

Employing Lemma 3, we obtain from (59) that

−τ

0



= x(t )T Qj x(t ) −

xt (s)T Q0 xt (s) ds



0

= x(t )T Qj x(t ) −

are given in Lemma 4 and Z0 (M − 1, M − 1) is where defined by (19), (21).



xt (s)T Q0 xt (s) ds −τ

while the derivatives of the terms of V2 corresponding to j ≥ 1 can be obtained as

(51)

Proof. We shall prove first the existence of a µ1 > 0 such that V (xt , x˙ t ) ≥ µ1 ∥x(t )∥. Consider the term of V2 with j = 0. Applying estimation (27) with ν0 = ν1,0 ≤ M − 1 and Ξ0 = I we obtain

0

= x(t )T Q0 x(t ) − x(t − τ )T Q0 x(t − τ ),

(49) (50)



d

(57)



˙ M (t )T Ψm3,1,M (τ ) − Ψm3,2,M (τ ) Φ M (t ). V 3 (t ) ≤ Φ 2 2

(65)

The statement of the theorem follows from (42), (57), (61) and (65) using the standard Lyapunov–Krasovskii Theorem (see e.g. [2]).  Remark 4. If Ad2 = 0 is considered in (6), Theorem 1 with M = N , m1 = 0, m2 = 1 gives back Theorem 5 of [4] (apart from a multiplier τ (i.e. h) in the derivative of V3 .) Remark 5 (Delay Range Stability). An analogous stability result can be proven, if Ad2 = 0 and τ is supposed to be an unknown constant, but for which a lower and an upper bound is known, i.e. τ ≤ τ ≤ τ

É. Gyurkovics, T. Takács / Systems & Control Letters 96 (2016) 72–80

for some given τ and τ . One can see that Ψm01 ,M (τ ) is affine in τ ,

(τ ) > 0 holds true for all τ ∈ [τ , τ ], provided that Ψm01 ,M (τ ) > 0. Moreover Ψm21 ,M does not depend on τ , while in this case, ΨM1 (τ ) is affine in τ , as well. One can modify the definition (41) of V3 (˙xt ) by taking the multiplier τ 2 in front of the summation 3,2 instead of τ , then τ −1 disappears from Ψm2 ,M (τ ). Apply Schur

and

Ψm01 ,M

3 ,1

complements to (42) with respect to the new Ψm2 ,M (τ ) and a congruence transformation, then we obtain

 3 ,2 1 2 ΨM (τ ) + Ψm1 ,M − Ψm2 ,M  Ψ (τ ) =   ∗

We show first that inequality Ψm0,M (τ ) > 0 implies Ψm0,M +1 (τ )

> 0 independently of ε > 0. In fact, matrix Ξˆ j is obtained by adding a new row and a new column to Ξj , i.e  Ξj ˆ Ξj = j,1 ξ M −j where ξ j

M −j

0



ξMj −j,M , is partitioned as ξ j

M −j

  = ξ jM,1−j , ξMj −j,M . Due to the (j)

definition of Pˆ in (70), the block-diagonal structure of Qj and the

m2

  Rj  τ AT  j=1  < 0. m2   − Rj

77

ˆ j , we obtain by standard algebra structure of Ξ (66)

Ψm0,M +1 (τ ) =



Ψm0,M (τ )



0 + PSDTs, εI

0

j =1

(71)

The matrix valued function Ψ (τ ) is affine in τ , which means that it is enough to require the fulfillment of the inequality (66) at the endpoints, i.e. the LMIs Ψm01 ,M (τ ) > 0, Ψ (τ ) < 0 and Ψ (τ ) < 0 have to hold true.

where PSDTs stand for positive semidefinite terms, therefore the statement follows.

4. Hierarchy of the LMI stability conditions

ΨM1 +1 (τ ) = ΓMT +1 Pˆ ΛM +1 + ΛTM +1 Pˆ ΓM +1 , with   ΛM 0 , ΓM +1 = diag {ΓM , τ I } , ΛM +1 = 0,1 0,2 ζM ⊗ I ζM I   where ζ 0 is again partitioned as ζ 0 = ζ 0,1 , ζ 0,2 with ζ 0,2 = M M M M M

This section is devoted to the comparison of the stability conditions obtained in the previous section for different parameters. We observe that parameter M determines the size of matrices P 0 , the number of columns of Ξj and Zk , while the number and L of rows of Ξj and Zk is ν1,j and ν2,k . The number of matrices Qj s and Rk s is m1 and m2 . The aim is to show that the LMI conditions can be arranged into a hierarchy table provided that the parameters are chosen to satisfy the following condition. M ≥ 1,

m1 = m − 1, ν1,j = M − j − 1, j = 0, . . . , m − 1,

m2 = m, ν2,k = M − k, k = 0, . . . , m − 1.

(67)

We shall refer to the LMI condition (42) with parameters satisfying (67) as Lm,M . Clearly, Lm,M depends on τ , too. We shall write Lm,M (τ ) when considering Lm,M for a given value of τ .

ˆ ) be given. We will ˆ,M Definition 1. Let the pairs (m, M ) and (m say that Lmˆ ,Mˆ outperforms Lm,M , if, for every τ for which Lm,M (τ ) has a feasible solution, Lmˆ ,Mˆ (τ ) has a feasible solution, too. This is denoted by Lm,M ≺ Lmˆ ,Mˆ . We will show that the parametric family of Lm,M is ordered according to both parameters. Theorem 2. Let the integer parameters satisfy (67). Then

Lm,M ≺ Lm,M +1 ,

(68)

Lm,M ≺ Lm+1,M .

(69)

Next we express ΨM1 +1 (τ ) by ΨM1 (τ ). By the definition of (τ ), we have

ΨM1 +1

−ζM0 ,M . Using (70), by standard algebra we obtain  1  ΨM (τ ) 0 ΨM1 +1 (τ ) = 0

+ ετ



0

0

0

ζ 0M,1 ⊗ I



 +

ζ 0M,2 I

 T  ζ 0M,1 ⊗ I

0

ζ 0M,2 I

0

.

(72)

Express now Ψm2,M +1 by Ψm2,M . Since

  Ξj−1 0 ˆ Ξj−1 = j−1,1 ξ M +1−j ξMj−+11−j,M ,   ˆ j(−j)1 = diag Qj(−j)1 , c2 (M , j)Qj , Q where c2 (M , j) = 2M − j + 2, we obtain  T   (j) ˆ j −1 ⊗ I Ξˆ j−1 ⊗ I Qˆ j−1 Ξ   T (j)   0 = Ξj−1 ⊗ I Qj−1 Ξj−1 ⊗ I + PSDTs, 0

0

therefore

Ψm2,M +1 ≤

 2 Ψm,M



0 . 0

(73)

Proof. Part 1. First we show (68). Let matrices P , Q0 , . . . , Qm−1 and R1 , . . . , Rm denote a feasible solution of Lm,M (τ ) for some fixed τ . We seek the solution of Lm,M +1 (τ ) in the form of

Express now Ψm,M +1 (τ ) by Ψm,M (τ ). Since Aˆ = (A, 0), we obtain

 ˆP = P

Ψm3,,1M +1 (τ ) =

0

Rˆ j = Rj ,



0 , εI

Qˆ i = Qi ,

(i = 0, . . . , m − 1),

(70)

(j = 1, . . . , m)

for some positive constant ε . In what follows, we shall denote matrices that belong to (M + 1, m) analogously by putting ‘‘hat’’ over them.

0

3,1



3,1

Ψm3,,1M (τ )



0 . 0

0

3,2

(74)

3,2

Express Ψm,M +1 (τ ) by Ψm,M (τ ). Since

 Zj−1 ˆ Zj−1 = ζ j−1,1 M +1−j

0

ζ

j−1,2 M +1−j



,

(j)



(j)



Rˆ j−1 = diag Rj−1 , c2 (M , j)Rj ,

78

É. Gyurkovics, T. Takács / Systems & Control Letters 96 (2016) 72–80





j − 1 ,1 = ζM , ζ jM−+1,12−j and ζ jM−+1,12−j = −ζMj−+11−j,M , we +1 −j

where ζ j−1

M +1−j

obtain

Ψm3,,2M +1

holds true. On the one hand,

Ψm3,+11,M (τ ) = τ AT

m

   1  ˆ j−1 ⊗ I T Rˆ j(−j)1 Zˆ j−1 ⊗ I = (τ ) = j Z τ

≥

τ



Zj−1 ⊗ I

j

T

(j) Rj−1



0

×

Zj−1 ⊗ I





0 + 0

0

j =1





ζ

0,1 M

−ζ

0

⊗I

T  

0 M ,M I

0 0

0 R1



0



0

ζ

0,1 M

−ζ

⊗I

=

c2 (M , 1)

,

(75)

Ψm3,,1M +1 (τ ) − Ψm3,,2M +1 (τ ) < 0. Applying (72)–(75) we obtain    T   Ψ m,M (τ ) 0 0 ,1 I ζM ⊗I 2M + 1 Ψ m,M +1 (τ ) ≤ 0 − R1 0 −ζM0 ,M I τ    T   0 0 ζ 0M,1 ⊗ I  I  × . (76) + ετ 0,1 ζM ⊗I −ζM0 ,M I 2ζ 0,2 I ζ 0,1 ⊗ I M

Since diag Ψ m,M (τ ), − 2Mτ+1 R1 < 0, and matrix





I





is non-singular, there exists a positive con-

0 −ζM ,M I

stant ν1 such that





I

0,1 ζM ⊗I

T   Ψ m,M (τ )

 ×

 I  0,1 ζM ⊗I



0

−

0

−ζM0 ,M I

0

2M + 1

τ

R1



0

−ζM0 ,M I

< −ν1 I ,

τ

ξ

0 0,1 M

⊗I

= Ψm3,,2M (τ ) +

m+1

τ

ε (Zm+1 ⊗ I )T Dm+1,M (Zm+1 ⊗ I ) .

(81)

It follows from (79)–(81) that

Ψ m+1,M (τ ) = Ψ m,M (τ ) + ε Ωm,M (τ ) with

Ωm,M (τ ) = E1T E1 − (m + 1) (Ξm ⊗ I )T Dm,M (Ξm ⊗ I ) m+1 + τ AT A − (Zm ⊗ I )T Dm+1,M (Zm ⊗ I ) . τ Then there exists a constant ν4 such that Ωm,M (τ ) ≤ ν4 I. If ε > 0 is small enough to satisfy inequality εν4 < ν3 , then Ψm+1,M (τ ) is negative. 

 T  0,1 ξM ⊗I 2ξ 0,2 I

u2,k ≤ ν2′ ,k ≤ M + 1 − k,

< ν2 I .

ν2′ ,k0 = M + 1 − k0 for some 0 < k0 ≤ m2 − 1.

If εν2 < ν1 , then Ψ m,M +1 (τ ) < 0 holds true. Part 2. Secondly we show that Lm,M ≺ Lm+1,M . First we show the positivity of Ψm0+1,M (τ ). Suppose that Ψm0,M (τ ) > 0 with the choice of P , Q0 , . . . , Qm−1 . We seek matrix Qm as Qm = ε I with ε > 0. Then we obtain

  ε diag 0, (Ξm ⊗ I )T Dm,M (Ξm ⊗ I ) , τ

(77)

Dm,M = diag{(m + 1)I , (m + 3)I , . . . , (2(M − m − 1)

+ m + 1)I }.

(78)

The first term of the right hand side of (77) is positive, the second term is non-negative, therefore Ψm0+1,M (τ ) > 0 for any ε > 0.

Next we show that Ψ m+1,M (τ ) < 0 has a feasible solution, provided that Ψ m,M (τ ) < 0 has also a feasible solution. Suppose that Ψ m,M (τ ) < −ν3 I with P , Q0 , . . . , Qm−1 and R0 , . . . , Rm−1 . We seek matrices Qm = ε I and Rm+1 = ε I, where ε > 0 has to be chosen. Matrix ΨM1 (τ ) is unchanged.   Denote E1 = I 0 . . . 0 ∈ Rnx ×nx (M +2) , then

Ψm2+1,M = Ψm2,M + E1T Qm E1

−(m + 1) (Ξm ⊗ I )T Qm (Ξm ⊗ I )

for all values of j, k,

(82)

and such that

M

Ψm0+1,M (τ ) = Ψm0,M (τ ) +

Remark 6. We note that (‘‘bidirectional’’ or ‘‘multidirectional’’) hierarchies of the LMI stability conditions can be established under weaker conditions than (67) at the cost of more involved notations. For example, let τ > 0, m1 ≥ 0, m2 ≥ 1 be fixed. Assume that LMI (42) with given M ≥ 1, ν1,0 , . . . , ν1,m1 , ν2,0 , . . . , ν2,m2 −1 satisfying conditions 0 ≤ ν1,j ≤ M − j − 1, 0 ≤ ν2,k ≤ M − k, has a feasible solution for τ . Consider now LMI (42) with M ′ = M + 1, ν1′ ,0 , . . . , ν1′ ,m1 , ν2′ ,0 , . . . , ν2′ ,m2 −1 satisfying conditions

ν1,j ≤ ν1′ ,j ≤ M − j,

and constant ν2 such that



(80)



0

ζ 0M,1 ⊗ I

(τ ) + ετ AT A,

Ψm3,+21,M (τ )

T T

where we omitted several positive semidefinite terms on the right hand side of (75). Finally we show that Ψ m,M +1 (τ ) = ΨM1 +1 (τ ) + Ψm2,M +1 +



Ψm3,,1M

while

τ

0 M ,M I

M

Rj A + τ AT Rm+1 A

j =1

j =1

m 1

m 

(79)

(83)

The corresponding LMI (42) has a feasible solution for the same τ , as well. In fact, condition (82) allows to get estimations in an analogous way as in the proof of Theorem 2. The only difference should be that in (75) we have to keep the term belonging to j = k0 instead of j = 1. Since c2 (M , k0 ) > 0, Rk0 > 0 and condition k −1

(83) ensures that ζM0+1−k0 ,M ̸= 0, the proof can be completed. We observe, however, that M , m1 , m2 determine the number of unknowns, therefore it is reasonable to apply Lemmas 3 and 4 with maximum ν1,j and ν2,k (i.e. use the tightest estimations within the given frames). Other cases can be investigated partly in a simpler, partly in an analogous way as in the proof of Theorem 2.

5. Numerical examples In this section, we apply the proposed method to three benchmark examples that have been extensively used in the literature to compare the results. The computations have been performed by using YALMIP [24] together with MATLAB. In all of these examples, Theorem 1 has been applied with m = m2 = m1 + 1, ν1,j = M − j − 1, ν2,j = M − j, (j = 0, 1, . . . , m − 1).

É. Gyurkovics, T. Takács / Systems & Control Letters 96 (2016) 72–80 Table 1 Systems used as illustrative examples. Example

A

 1

0

 2

 3

5.2. Discussion

Ad1

−2 0.2 0.2 0

−2

  −1 −1  

Ad2

0 −0.9 0 0.1

1 0.1

0 0





0 1

0 −1





0 0

−1 −1

0 0





0 0





0 0

Analytical bounds 0 0

τ =0 τ ∼ 6.17258

 

0 −1 0 0

τ ∼ 0.2 τ ∼ 2.04 τ ∼ 0.1002 τ ∼ 1.7178



Table 2 Delay bounds for Example 1 obtained by Theorem 1. m

1

1

1

1

4

M

1 6.05932 16

2 6.16894 27

3 6.172504 42

4 6.17258 61

3 6.17258 60

τ

NoDV

Table 3 Delay upper bounds for Example 2. Method

M

1

2

3

4

5

[14] Theorem 1, m = 1

τ τ

– 1.9419 16

1.58 2.0395 27

1.83 2.0412 42

1.95 2.0412 61

2.02 2.0412 84

NoDV

Table 4 Delay bounds for Example 3. Method [5]Theorem 1 Theorem 1, (m = 1)

79

M

τ

τ

NoDV

1 2 3

0.1002 0.10055 0.10018 0.10017

1.5954 1.5405 1.7122 1.71799

59 16 27 42

5.1. Numerical experiments Consider system (1) with coefficient matrices listed in Table 1. Example 1 is considered in numerous papers, among others, in [4,7,13], where extensive comparisons with previously reported results are given. The results obtained by Theorem 1 are given in Table 2. The results obtained with m = 1, M = 1; 2; 3 coincide with that of [4], which are the best previously reported results we are aware of. (See Remark 4.) We note that the same bounds of τ are obtained for m = 2 and m = 3 as for m = 1 with the same values of M, however an improvement is obtained in the case of m = 4, M = 3 compared to m = 1, M = 3. NoDV denotes in all tables the number of decision variables. Example 2 is considered in many papers, among others, in [5,12,14], where extensive comparisons with previously reported results are given. In [12], a delay bounding interval [0.200, 1.877] was obtained with 16 decision variables, while the authors of [5] derived the delay bounding interval [0.2000, 1.9504] from Theorem 1 with 59 decision variables. In [14], the lower bound of the delay was found to be 0.2001, while by Theorem 1, we obtained the lower bound 0.20001. The upper bounds reported in [14] and obtained by Theorem 1 are given in Table 3. The values of N in [14] and M in the present paper are related as N = M + 1. We note that the same upper bounds were obtained for different values of m for a given value of M . Example 3 is also widely used for comparing the effectiveness of different methods. Here we shall mention a recently published work in [5], where comparisons with previously reported results are given, as well (see also [15]). The results obtained by [5] and by Theorem 1 are given in Table 4. We note that the same upper bounds were obtained for different values of m for a given value of M for this example, too.

It can be seen that, in these examples, Theorem 1 yields better delay bounds than previously published methods except of Theorem 5 of [4] which is equivalent to Theorem 1, if it is applied with m1 = 0, m2 = 1. (This corresponds to the case of m = 0 in [4].) It is worth noting that these better results are obtained with much smaller number of decision variables. In these and in several other examples from the literature, on which we tested our approach, we observed that the improvement of the delay estimation is primarily due to the increase of the dimension of the extended state variable together with the improved lower bounds of Lemmas 3 and 4. This does not contradict to the reported improvements in the case of the application of triple, etc. integral terms in the LKF, since the applied lower estimations of the integrals of quadratic terms lead to the introduction of some extended state variables with increased dimension, as well. We emphasize that this remark is limited to the investigated examples, and the observed behavior may have the reason that the analytical bounds were rapidly reached up to 4–6 digits. Moreover, we note similarly to [4] that the formulated result does not establish any convergence to the analytical bounds. 6. Conclusion In this paper, new multiple integral inequalities are derived based on certain hypergeometric-type orthogonal polynomials. These inequalities are similar to that of [19], and they comprise the famous Jensen’s and Wirtinger’s inequalities, as well as the recently presented Bessel–Legendre inequalities of [4] and the Wirtinger-based multiple-integral inequalities of [5,6]. Applying the obtained inequalities, a set of sufficient LMI stability conditions for linear continuous-time delay systems are derived. It was proven that these LMI conditions could be arranged into a bidirectional hierarchy establishing a rigorous theoretical basis for comparison of conservatism of the investigated methods. Numerical examples confirm that the proposed method enhances the tolerable delay bounds. Acknowledgments The authors would like to thank the anonymous reviewers for their constructive comments that have greatly improved the quality of this paper. References [1] C. Briat, Linear Parameter-varying and Time-delay Systems, Springer, 2014. [2] E. Fridman, Introduction to Time-delay Systems: Analysis and Control, Springer, 2014. [3] M. Wu, M, Y. He, J.-H. She, Stability Analysis and Robust Control of Time-Delay Systems, Science Press, Beijing and Springer-Verlag, Berlin, Heidelberg, 2010. [4] A. Seuret, F. Gouaisbaut, Hierarchy of LMI conditions for the stability analysis of time delay systems, Systems Control Lett. 81 (2015) 1–7. [5] M. Park, O.M. Kwon, J.H. Park, S.M. Lee, E. Cha, Stability of time-delay systems via Wirtinger-based integral inequality, Automatica 55 (2015) 204–208. [6] T.H. Lee, J.G. Park, M.J. Park, O.M. Kwon, H.-Y. Jung, On stability criteria for neural networks with time-varying delay using Wirtinger-based multiple integral inequality, J. Franklin Inst. B 352 (2015) 5627–5645. [7] J.-H. Kim, Further improvement of Jensen inequality and application to stability if time-delayed systems, Automatica 64 (2016) 121–125. [8] O.M. Kwon, M.J. Park, J.H. Park, S.M. Lee, E.J. Cha, On Less Conservative Stability Criteria for Neural Networks with Time-varying Delays Utilizing Wirtinger-based Integral Inequality, in: Mathematical Problems in Engineering, 2014, Hindawi Publishing Corporation, 2014, http://dx.doi.org/10.1155/2014/859736, ID 859736. [9] K. Liu, E. Fridman, Wirtinger’s inequality and Lyapunov-based sampled-data stabilization, Automatica 48 (2012) 102–108. [10] P. Park, Won Il Lee, S.Y. Lee, Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delay systems, J. Franklin Inst. B 352 (2015) 1378–1396.

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