Generalized integral inequality: Application to time-delay systems

Accepted Manuscript Generalized integral inequality: Application to time-delay systems

M.J. Park, O.M. Kwon, J.H. Ryu

PII: DOI: Reference:

S0893-9659(17)30289-6 https://doi.org/10.1016/j.aml.2017.09.010 AML 5338

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Applied Mathematics Letters

Received date : 18 August 2017 Accepted date : 19 September 2017 Please cite this article as: M.J. Park, O.M. Kwon, J.H. Ryu, Generalized integral inequality: Application to time-delay systems, Appl. Math. Lett. (2017), https://doi.org/10.1016/j.aml.2017.09.010 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.

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Generalized integral inequality: Application to time-delay systems M.J. Parka , O.M. Kwonb,∗, J.H. Ryuc,∗ a Center

for Global Converging Humanities, Kyung Hee University, 1732 Deogyeong-daero, Giheung-gu, Yongin 17104, Republic of Korea b School of Electrical Engineering, Chungbuk National University, 1 Chungdae-ro, Seowon-gu, Cheongju 28644, Republic of Korea c Electronics and Telecommunications Research Institute, 176-11 Cheomdan Gwagi-ro, Buk-gu, Gwangju 61012, Republic of Korea.

Abstract This paper investigates a stability problem for linear systems with time-delay. By constructing simple Lyapunov-Krasovskii functional (LKF), and utilizing a new generalized integral inequality (GII) proposed in this paper, a sufficient stability condition for the systems will be derived in terms of linear matrix inequalities (LMIs). Two illustrative examples are given to show the superiorities of the proposed criterion. Keywords: Systems with time-delays, time-invariant, generalized integral inequality, Lyapunov method.

1. Introduction Let us consider time-delay systems (TDSs) given by x(t) ˙ x(t)

= =

Ax(t) + Ad x(t − h) + AD φ(t), t ∈ [−h, 0],

Z

t

x(s)ds,

t−h

(1)

where x(t) ∈ Rn is the state vector, φ(t) is an initial function, A, Ad and AD ∈ Rn×n are known constant matrices, and h ∈ [hm , hM ] is a time-constant delay. Here, because it is well known that the time-delay causes poor performance or instability of systems, a great number of results on delay-dependent stability condition for TDSs have been reported in the literature. In this field, an important issue is to find less conservative conditions guaranteeing the asymptotic stability of TDSs. Here, the conditions are classified under two main perspectives that they are referred to Section 4 for further details. Naturally, various methods exist but the primary concern is a fundamental study on a new bound of the inequality for the integral of quadratic functions (IQFs). In this regards, many researchers have put their times and efforts on developments of IQFs such as the free matrix-based multiple integral inequality [1], Jensen integral inequality (JII) [2], Wirtinger-based integral inequality (WBI) [10], the different varieties of Wirtinger-based double integral inequality [5, 12, 13], Bessel-Legendre inequality (BLI) [11], and the auxiliary function-based integral inequalities (AFBIIs) [8, 9]. What all authors are noticing is how tightly bounded the inequality for IQFs is. Moreover, Cauchy-Schwartz-like inequality was addressed in the authors’ works [6, 7] as summation case. Even though the aforementioned inequalities have different types of these, most of them can be derived from GII. In line with this thinking, by applying Gram-Schmidt orthogonalization process and considering an weighted function unlike the work [9], GII is established in form of the inequality for the k + 1 tuple IQFs RbRb Rb given by a s1 · · · sk f (u)dudsk · · · ds1 , where f (t) is any quadratic function. The main advantage of the ∗ Corresponding

authors. Email addresses: [email protected] (M.J. Park), [email protected] (O.M. Kwon), [email protected] (J.H. Ryu)

Preprint submitted to Applied Mathematics Letters

August 18, 2017

proposed GII is that GII embraces most cases in the existing works. By utilizing the GII and constructing a simple LKF, a stability criterion for system (1) which provides larger delay bounds with fewer decision variables is derived in Theorem 1. Finally, advantages of employing the proposed inequality are illustrated via two numerical examples. Notation. Throughout this paper, the used notations are standard. Rn and Rm×n denote the n-dimensional Euclidean space with the Euclidean vector norm k · k and the set of m × n real matrices, respectively. Sn+ is the set of symmetric positive definite n × n matrices. X > 0 (< 0) means symmetric positive (negative) definite matrix. X⊥ denotes a basis for the null-space of X. diag{· · ·}, and He{X} stand for, respectively, the (block) diagonal matrix, the sum of X and X T . 2. Generalized integral inequality The main objective of this section is to propose GII to analyze the stability for the system (1). The following fact is utilized to derive Lemma 1 presented about GII. Fact 1 (Gram-Schmidt orthogonalization [3]). Let V be a vector space with an inner product < ·, · >. Suppose e1 , e2 , . . ., en , is a basis for V . Let p1 = e 1 , pi = e i −

i−1 X < e i , pj > pj , i = 2, 3, . . . , n. < pj , pj > j=1

Then p1 , p2 , . . ., pn is an orthogonal basis for V . Lemma 1. For a matrix M ∈ Sn+ , scalars a < b, non-negative integer k, l and vector z : [a, b] → Rn , the following inequality holds Z

a

b

wk (s)z T (s)M z(s)ds ≥

Z

a

b

Pl,k (s) ⊗ z(s)ds

T

Z (Ψl,k ⊗ M )

a

b

 Pl,k (s) ⊗ z(s)ds ,

(2)

where for i = 0, 1, . . . , l, wk (s) = (s − a)k , ei (s) = (s − a)i , pi,k (s) = ei (s) − ψj,k (ei (u)) =

Rb

a ei (u)wk (u)pj−1,k (u)du , Rb w (u)p2j−1,k (u)du a k

 Ψl,k = diag R b a

1

wk (s)p20,k (s)ds

,..., Rb a

i X

ψj,k (ei (u))pj−1,k (s),

j=1

Pl,k (s) = wk (s)col{p0,k (s), . . . , pl,k (s)} 1

wk (s)p2l,k (s)ds

 .

Proof. Defining a vector function g : [a, b] → Rn :

  l+1 X p 1 ψj,k (z(u))pj−1,k (s) , g(s) = wk (s)M 2 z(s) − j=1

1

where M 2 means the principal square root of the matrix M , and considering the orthogonality of pi,k (s) based on Fact 1 lead to

  2 Z b Z b l+1 X

1 2

ψj,k (z(u))pj−1 (s) kg(s)k ds = wk (s) M 2 z(s) −

ds a

a

j=1

2

Z

=

b

wk (s)z (s)M z(s)ds − 2

a

+

T

Z

b

a

Z

=

b

a

+ Z

a

X l+1 j=1

 ψj,k (z(u))pj−1,k (s) ds

j=1

wk (s)z T (s)M z(s)ds − Rb

1

l+1 X

φTj,k (s)M φj,k (u) 2 Rb 2 j=1 a wk (u)pj−1,k (u)du

2 a wk (u)pj−1,k (u)du

wk (s)z T (s)M z(s)ds −

b

a

wk (s)z (s)M

a

l+1 X j=1

Rb where φj,k (α) = a z(α)wk (α)pj−1,k (α)dα. Rb Since a kg(s)k2 ds ≥ 0, the following inequality is valid: Z

T

j=1

l+1  X b

b

 T  X X l+1 l+1 ψj,k (z(u))pj−1,k (s) ds ψj,k (z(u))pj−1,k (s) M wk (s)

j=1

=

Z

wk (s)z T (s)M z(s)ds ≥

l+1 X j=1

Hence, the above inequality can be rearranged as (2).

2 Z

a

b

 wk (s)p2j−1,k (s)ds φTj,k (u)M φj,k (u)

φTj,k (s)M φj,k (s)

Rb

2 a wk (s)pj−1,k (s)ds

φTj,k (s)M φj,k (s)

Rb a

wk (s)p2j−1,k (s)ds

,

.

(3)

(4) 

The following remark gives an explanation of meanings of k and l. Specially, the tightness of the inequality for the k + 1 tuple IQFs increases as the value of l increases and follows for the single IQFs Rb Rb f (s)ds (k = 0): a f (s)ds ≥ F0 + F1 + · · · + Fl ≥ · · · ≥ F0 + F1 ≥ F0 ,where Fl is a constant term to be a obtained by Lemma 1. Then, the l shall be called “degree of tightness”. Remark 1. The inequality proposed by Lemma 1 is presented by considering the work [9] and the k + 1 tuple integral of f (t) given by Z b Z b Z bZ b wk (u) f (u)dudsk · · · ds1 = ··· f (u)du. (5) k! a sk a s1 Here is what it means: the function pi,k (s) inspired from the work [9] is weighted by wk (s) = (s − a)k to reflect the relationship (5) to obtain the inequality for the k + 1 tuple IQFs. Then, the inequality proposed by Lemma 1 can be generalized for various inequalities including JII [2], AFBIIs [8, 9], WBI [10], and BLI [11] according to the selection of the values l and k. The following integral functions: Z b Z Z (b − a)k+1 b b Φk (z) = f (u)dudsk · · · ds1 , ··· (k + 1)! a s1 sk where f (u) = z T (u)M z(u), can be bounded as follows: • When l = 0, the proposed inequality (2) can be obtained as JII for the k + 1 tuple IQFs: where Σk (z) =

RbRb a

s1

···

Rb

sk

Φk (z) ≥ ΣTk (z)M Σk (z),

(6)

z(s)dsdsk · · · ds1 . In addition, selecting k = 0 leads to Lemma 1 in [2].

• When l = 1, the inequality (2) can be calculated as WBI for the k + 1 tuple IQFs: b Tk (z)M Σ b k (z), Φk (z) ≥ ΣTk (z)M Σk (z) + (k + 1)(k + 3)Σ

b k (z) = Σk (z) − where Σ

k+2 b−a Σk+1 (z).

In addition, selecting k = 0 leads to Corollary 4 in [10]. 3

(7)

Table 1: Compared with the existing works.

Methods Lemma 5.1 [8] Lemma 3 [9] Corollary 4 [11] Lemma 2.3 [13] Ours

k ≤1 0 0 1 ≥0

l ≤2 ≤3 ≥0 2 ≥0

• The case of l = 2 corresponds to AFBII for the k + 1 tuple IQFs. In addition, selecting k = 0 leads to Remark 4 in [8]. Park et al. [9] worked for k = 0 and l ≥ 3. However, Lemma 1 in this work supports the inequality for the k + 1 tuple IQFs. • When k = 0, the inequality (2) can be derived as BLI for the single IQFs: Φ0 (z) ≥

l X e T (z)M Σ e i (z), (2i + 1)Σ i

(8)

i=0

where the Legendre orthogonal polynomials Li (s, b−a) = (−1)i R e i (z) = b Li (s, b − a)z(s)ds. and Σ a

Pi

i s−a j j=0 qj ( b−a )

(i+j)! with qji = (−1)j (i−j)!(j!) 2

As a result, Lemma 1 marks with a side dot for the existing inequality applications to time-delay systems. Remark 2. In addition to Remark 1, the integral inequalities proposed in the existing works [8, 9, 11, 13] are special cases of Lemma 1 established in this work. Their details are summarized as Table 1 by substituting z(s) ˙ into z(s). Remark 3. In this paper, to trade off between the improved result and the computational burden, the values of l and k are selected by 3 and 0, respectively; that is, Φ0 (z) ≥

2 X

(2i + 1)ΓTi (z)M Γi (z),

(9)

i=0

Rb R bR bR b R bR b Rb 2 12 where Γ0 (z) = a z(s)ds, Γ1 (z) = b−a z(u)duds − a z(s)ds, Γ2 (z) = (b−a) 2 a s u z(v)dvduds − a s R bR b R bR bR bR b R bR bR b R bR b Rb 120 12 60 6 b−a a s z(u)duds+ a z(s)ds, Γ3 (z) = (b−a)3 a s u v z(r)drdvduds− (b−a)2 a s u z(v)dvduds+ b−a a s z(u)duds− Rb a z(s)ds. For the details, from (2) with l = 3 and k = 0, the above inequality can be obtained by     1 p0,0 (s) b−a   p1,0 (s)   (s − a) − 2 ,   (10) P3,0 (s) = w0 (s)  (b−a)2   p2,0 (s)  =  2 (s − a) − (b − a)(s − a) + 6 6 1 p3,0 (s) (b − a)2 (s − a) − 20 (b − a)3 (s − a)3 − 32 (b − a)(s − a)2 + 10 and

Ψ3,0

 = diag R b

2 a w0 (s)p0,0 (s)ds

Note that =

(b−a) 2800

1

7

Rb a

,..., Rb

w0 (s)p20,0 (s)ds = b−a,

1

2 a w0 (s)p3,0 (s)ds

Rb a



w0 (s)p21,0 (s)ds =

  1 180 2800 12 = diag . (11) , , , b − a (b − a)3 (b − a)5 (b − a)7

(b−a)3 12 ,

Rb a

w0 (s)p22,0 (s)ds =

(b−a)5 180 ,

and

. Then, by using the above matrices, the following relationship can be calculated: Z b Z b z T (s)M z(s)ds = w0 (s)z T (s)M z(s)ds a

a

4

Rb a

w0 (s)p23,0 (s)ds

≥

=

Z

a

b

P3,0 (s) ⊗ z(s)ds

T

Z b  (Ψ3,0 ⊗ M ) P3,0 (s) ⊗ z(s)ds

T  Γ0 (z)   Γ1 (z)   b    Γ2 (z)  (Ψ3,0 ⊗ M )  Γ3 (z) 

 Γ0 (z) Γ1 (z)  , Γ2 (z)  Γ3 (z)

a

(12)

R bR bR bR b R bR bR b R bR b 2R b z(s)ds, ∆2 = 6 a v u s z(r)drdvduds − where ∆1 = 2 a s u z(v)dvduds − (b − a) a s z(u)duds + (b−a) 6 a R bR bR b R bR b Rb 6 1 b 3,0 = diag{ 1 , 3 , 5 , 7 }, 3(b − a) a u s z(v)dvduds + 10 (b − a)2 a s z(u)duds − 20 (b − a)3 a z(s)ds, Ψ b−a b−a b−a b−a and Γi (z) (i = 0, 1, 2, 3) were defined in (9). Finally, by multiplying both sides of (12) by b − a, the inequality (9) can be easily obtained. 3. Stability analysis In this section, based on Lemma 1 and Remark 3, an improved stability condition is derived as follows. Theorem 1. For a given positive scalar h ∈ [hm , hM ], the system (1) is asymptotically stable, if there n n exist matrices P ∈ S4n + , Q ∈ S+ , R ∈ S+ satisfying the following LMI: ΥT⊥ (Ξ1 − Ξ2 )Υ⊥ < 0,

(13)

k

60 120 where θk = hk! e1 −e3+k , π1 = e1 −e2 , π2 = −π1 + h2 θ1 , π3 = π1 − h6 θ1 + h122 θ2 , π4 = −π1 + 12 h θ1 − h 2 θ2 + h 3 θ3 , P4 T T T T 2 T Ξ1 = He{[e1 , e4 , e5 , e6 ]P [e3 , π1 , θ1 , θ2 ] }+e1 Qe1 −e2 Qe2 +h e3 Re3 , Ξ2 = k=1 (2k −1)πk Rπk , Υ = AeT1 + Ad eT2 − eT3 + AD eT4 , and ei ∈ R6n×n (i = 1, 2, . . . , 6) means the block entry matrices; e.g., eT2 ζ(t) = x(t − h) h iT Rt Rt Rt R t R tR t with ζ(t) = xT (t), xT (t − h), x˙ T (t), t−h xT (s)ds, t−h s xT (u)duds, t−h s u xT (v)dvduds . Proof. Consider the following LKF candidate as Z t Z t Z t V = ν T (t)P ν(t) + xT (s)Qx(s)ds + h x˙ T (u)Rx(u)duds, ˙ (14) t−h

t−h s

where ν(t) = [e1 , e4 , e5 , e6 ]T ζ(t). By Lemma 1, the time-derivative of V is bounded as Z t V˙ = ζ T (t)Ξ1 ζ(t) − h x˙ T (s)Rx(s)ds ˙ ≤ ζ T (t)(Ξ1 − Ξ2 )ζ(t)

(15)

t−h

Then, a new stability condition for the system (1) can be Ξ1 − Ξ2 < 0

s.t. Υζ(t) = 0.

(16)

Finally, by Finsler’s Lemma in [4], if the LMI (13) holds then the condition (16) is satisfied, which means that the system (1) is asymptotically stable.  4. Examples Consider the systems listed in Table 2. In checking the conservatism of delay-dependent stability criteria, an important index is not only to obtain maximum delay bounds for guaranteeing asymptotic stability of time-delay systems as large as possible but also to use the decision variables while keeping the same maximum delay bound as few as possible. In this regards, Table 3 shows the results of the maximum delay bounds hM (or ranges [hm , hM ]) and the number of decision variables (#DV). It can be seen that Theorem 1 provides competitive results with the most criteria from the existing works. 5

Table 2: Systems used as Example.

No.

Systems     −2 0 −1 0 x(t) ˙ = x(t) + x(t − h)  0 −0.9  −1 −1 Rt 0.2 0 −1 0 x(t) ˙ = x(t) + x(s)ds t−h 0.2 0.1 −1 −1

1 2

Table 3: Results for each system.

No. 1

2

Methods [13] [12] [2] (N = 3) [1] Theorem 1 Analytical bound [12] [13] [1] Theorem 1

Results 6.1663 6.1664 6.171 6.1719 6.1719 6.1725 [0.2000, 2.0395] [0.2000, 2.0402] [0.2000, 2.0412] [0.2000, 2.0412]

#DV 45 75 67 106 42 75 45 106 42

5. Conclusion In this work, the generalized integral inequality for the multiple integral of quadratic functions which covers most of existing inequalities was proposed in Lemma 1. Based on the result of Lemma 1, the delaydependent stability criterion was derived in terms of LMI. Two numerical examples have been given to show the usefulness of the presented criterion. References [1] J. Chen, S. Xu, B. Zhang, Single/Multiple Integral Inequalities With Applications to Stability Analysis of Time-Delay Systems, IEEE Transactions on Automatic Control 62(7) (2017) 3488-3493. [2] K. Gu, V.L. Kharitonov, J. Chen, Stability of time-delay systems, Birkh¨ auser 2003. [3] W. Ledermann, S. Vajda, Handbook of Applicable Mathematics Vol. IV: Analysis, John Wiley & Sons 1980. [4] M.C. de Oliveira, R.E. Skelton, Stability tests for constrained linear systems, Springer-Verlag 2003. [5] M.J. Park, O.M. Kwon, J.H. Park, S.M. Lee, E.J. Cha, Stability of time-delay systems via Wirtinger-based double integral inequality, Automatica 55 (2015) 204-208. [6] M.J. Park, O.M. Kwon, Stability and Stabilization of Discrete-Time T-S Fuzzy Systems With Time-Varying Delay via Cauchy-Schwartz-Based Summation Inequality, IEEE Transations on Fuzzy Systems 25(1) (2017) 128-140. [7] 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 Analysis: Hybrid Systems 23 (2017) 76-90. [8] P.G. Park, W.I. Lee, S.Y. Lee, Auxiliary function-based integral inequalities for quadratic functions and their applications to time-delaysystems, Journal of the Franklin Institute 352 (2015) 1378-1396. [9] P.G. Park, W.I. Lee, S.Y. Lee, Auxiliary Function-based Integral/Summation Inequalities: Application to Continuous/Discrete Time-Delay Systems, International Journal of Control, Automation and Systems 14(1) (2016) 3-11. [10] A. Seuret, F. Gouaisbaut, Wirtinger-based integral inequality: Application to time-delay systems, Automatica 49 (2013) 2860-2866. [11] A. Seuret, F. Gouaisbaut, Hierarchy of LMI conditions for the stability analysis of time-delay systems, Systems & Control Letters 81 (2015) 1-8. [12] H.B. Zeng, Y. He, M. Wu, J. She, New results on stability analysis for systems with discrete delay, Automatica 60 (2015) 189-192. [13] N. Zhao, C. Lin, B. Chen, Q.-C. Wang, A new double integral inequality and application to stability test for time-delay systems, Applied Mathematics Letters 65 (2017) 26-31.

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