Automatica 46 (2010) 610–614

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Automatica

journal homepage: www.elsevier.com/locate/automatica

Technical communique

A delay-partitioning approach to the stability analysis of discrete-time systemsI

Xiangyu Meng a,b,∗, James Lam b, Baozhu Du b, Huijun Gao aa Department of Control Science and Engineering, Harbin Institute of Technology, Harbin 150001, Chinab Department of Mechanical Engineering, University of Hong Kong, Hong Kong

a r t i c l e i n f o

Article history:Received in revised form23 August 2009Accepted 18 November 2009Available online 16 December 2009

Keywords:Asymptotic stabilityDiscrete-time systemsDelay systems

a b s t r a c t

This paper revisits the problem of stability analysis for linear discrete-time systems with time-varyingdelay in the state. By utilizing the delay partitioning idea, new stability criteria are proposed in terms oflinear matrix inequalities (LMIs). These conditions are developed based on a novel Lyapunov functional.In addition to delay dependence, the obtained conditions are also dependent on the partitioning size. Wehave also established that the conservatism of the conditions is a non-increasing function of the numberof partitions. Numerical examples are given to illustrate the effectiveness and advantage of the proposedmethods.

© 2009 Elsevier Ltd. All rights reserved.

1. Introduction

As an important and fundamental problem, stability analysishas been at the forefront of the research on time-delay systemsin recent years. Compared with continuous-time systems withtime delays (Papachristodoulou, 2004; Peet, Papachristodoulou, &Lall, 2009), discrete-time systems with state delay have a strongbackground in engineering applications, among which network-based control has been well recognized to be a typical example.However, little effort has been made towards investigating thestability of discrete time-delay systems (Chen, Guan, & Lu, 2003).So far, a few approaches have been proposed to solve discrete-time systems with time delay. For a constant delay, a delay systemcan be converted to a delay-free one by using the so-called liftingor state-augmentation approach (Xu, Lam, & Zhang, 2002), whilesystems with time-varying delays have been transformed intoswitched systems in Hetel, Daafouz, and Iung (2008) and Xia,Liu, Shi, Rees, and Thomas (2007), so that classic results can beapplied to analyze the problems of stability. Solving the problem ofstability without performing model transformation to the originalsystem is another research direction. In Lee and Kwon (2002),a delay-dependent stability condition is presented for discrete-time systems with unknown constant delay. Improved delay-dependent conditions are provided in Xu, Lam, and Zou (2005),

I This paper was not presented at any IFAC meeting. This paper wasrecommended for publication in revised form by Associate Editor Emilia Fridmanunder the direction of Editor André L. Tits.∗ Corresponding address: University of Alberta, 9107 - 116 Street 2nd Floor ECERFBldg., T6G 2V4 Edmonton, Canada. Tel.: +1 780 4924875; fax: +1 780 4921811.E-mail addresses: xmeng2@ece.ualberta.ca (X. Meng), james.lam@hku.hk

(J. Lam).

0005-1098/$ – see front matter© 2009 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2009.12.004

which has been established that the proposed conditions areless conservative. For time-varying delay, a stability condition isproposed in Gao, Lam, Wang, and Wang (2004) by using Moon’sinequality (Moon, Park, Kwon, & Lee, 2001), which is dependent onthe minimum and maximum delay bounds. However, some usefulterms are ignored in Gao et al. (2004); themethod in Gao and Chen(2007) improved the result in Gao et al. (2004) by defining newLyapunov functions and by using bounding inequalities for crossproducts between two vectors. When revisiting this problem, wefind that the results reported in the literature still leavemuch roomfor improvement. The choices of specific Lyapunov functionalsand bounding techniques are the origin of conservatism. InGouaisbaut and Peaucelle (2006), artificial fractioning of the delayis introduced to give a sequence of LMI conditions for continuous-time systems. The same idea is also introduced to study thestability of continuous systems with multiple time-varying delaycomponents (Du, Lam, Shu, & Wang, 2009). Another relatedapproach can be found in Gu, Kharitonov, and Chen (2003), inwhich the discretized functional is used. This motivated us to carryout the present study.In this paper, for the first time, we utilize the delay partitioning

idea to solve the problem of stability analysis for linear discretesystemswith time-varying delay in the state. The aim of this paperis to provide tractable conditions for stability analysis, which havesignificantly reduced conservatism. This reduced conservatismbenefits from the fact that the free-weighting matrix approach(Wu, He, She, & Liu, 2004) is employed and the delay partitioningidea is adopted. In addition to delay dependence, the obtainedconditions are also dependent on the partitioning part. Theapproach developed has much lower computational complexitythan those using full state augmentation, and our method is

X. Meng et al. / Automatica 46 (2010) 610–614 611

more general: it covers the complete-augmentation and the no-augmentation cases. Numerical examples are given to illustrate theeffectiveness and advantage of the proposed methods.

Notation: The notation throughout the paper is standard. Thenotation P > 0 ( ≥ 0) means that P is real symmetric and positivedefinite (semi-definite). A symmetric term in a symmetric matrixis denoted by ∗, diag{. . .} stands for a block-diagonal matrix, and‖ · ‖ denotes the Euclidean norm. In addition, In and 0m×n denotethe n× n identity matrix andm× n zero matrix, respectively.

2. Problem formulation and preliminaries

Consider the following linear discrete-time system with time-varying delay in the state:

x (k+ 1) = Ax(k)+ Bx (k− d(k)) ,

x(k) = φ(k), k = −d,−d+ 1, . . . , 0. (1)

Here x(k) ∈ Rn is the system state vector. A and B are constant ma-trices with appropriate dimensions; the time delay d(k) is a posi-tive integer, is assumed to be time-varying in the whole dynamicprocess, and satisfies 1 ≤ d ≤ d(k) ≤ d, where d and d are constantpositive scalars representing the minimum and maximum delays,respectively. {φ(k), k = −d,−d+ 1, . . . , 0} is a known given ini-tial condition sequence. The lower bound of the delay d can alwaysbe described by d = τm, where τ andm are integers.We first represent the time delay d(k) in two parts: the constant

part τm and the time-varying part h(k); that is, d(k) = τm+ h(k),where h(k) satisfies 0 ≤ h(k) ≤ d− τm. Define

Υ (i) ,[xT (i) xT (i− τ) · · · xT (i− τm+ τ)

]T.

Definition 1. The origin of the above system is said to be asymp-totically stable, if, for any ε > 0, there exists δ > 0 such that, if‖φ(k)‖ < δ, k = −d,−d + 1, . . . , 0, then ‖x(k)‖ < ε, for everyk ≥ 0 and limk→∞ x(k) = 0.

Then, by applying the delay partitioning idea to the lower delaybound d = τm to give m parts, we construct the following newLyapunov functional candidate as

V (k) , V1(k)+ V2 (k)+ V3(k)+ V4(k), (2)

with

V1(k) , xT (k)Px(k),

V2(k) ,k−1∑i=k−τ

Υ T (i)Q1Υ (i)+k−1∑i=k−d

xT (i)Q2x (i) ,

V3(k) ,−τm+1∑j=−d+1

k−1∑i=k−1+j

xT (i) Rx (i) ,

V4(k) ,−1∑i=−τ

k−1∑j=k+i

δT (j) S1δ (j)+−τm−1∑i=−d

k−1∑j=k+i

δT (j) S2δ (j) ,

δ (j) , x (j+ 1)− x (j) ,

where P > 0, Qi > 0, i = 1, 2, R > 0, and Sj > 0, j = 1, 2 are to bedetermined.

3. Main results

This section is concerned with the stability analysis problem,and the main result is as follows.

Theorem 2. Given positive integers τ ,m, and d, the system in (1) isasymptotically stable if there exist real matrices P > 0, Qi > 0,i = 1, 2, R > 0, Sj > 0, j = 1, 2, M ≥ 0, N ≥ 0, X, Y and Zsatisfying[Φ + Ψ + Ψ T Ξ

∗ −diag {P,Q2, R, S1, S2}

]< 0, (3)

Π1 =

[M X∗ S1

]≥ 0, Π2 =

[N Y∗ S2

]≥ 0, (4)

Π3 =

[N Z∗ S2

]≥ 0, (5)

where

Φ = −Ξ T2 PΞ2 +WTQ1 Q1WQ1 −W

TR RWR −WQ2Q2WQ2

+ τM +(d− τm

)N,

Ψ =[X Y Z

] In −In 0n×(m+1)n0n×mn In −In 0n0n×(m+1)n In −In

,Q1 =

[Q1 00 −Q1

], WQ1 =

[Imn 0mn×3n

0mn×n Imn 0mn×2n

],

Ξ1 =[A 0n×mn B 0n

], WQ2 =

[0n×(m+2)n In

],

Ξ2 =[In 0n×(m+2)n

], WR =

[0n×(m+1)n In 0n

],

Ξ =

[Ξ T1 P Ξ T2 Q2

√d− τm+ 1Ξ T2 R

√τΞ T3 S1√

d− τmΞ T3 S2], Ξ3 = Ξ1 − Ξ2.

Proof. Taking the forwarddifference of the functional (2) along thesolution of system (1), we have

1V1(k) = ζ T (k)Ξ T1 PΞ1ζ (k)− xT (k)Px(k),

1V2(k) = ζ T (k)Ξ T2 Q2Ξ2ζ (k)− xT (k− d)Q2x (k− d)

+Υ T (k)Q1Υ (k)− Υ (k− τ)T Q1Υ (k− τ) ,

1V3(k) ≤(d− τm+ 1

)ζ T (k)Ξ T2 RΞ2ζ

T (k)− xT (k− d(k)) Rx (k− d(k)) ,

1V4(k) = ζ T (k)Ξ T3(τS1 +

(d− τm

)S2)Ξ3ζ (k)

−

k−1∑j=k−τ

δT (j) S1δ (j)−k−τm−1∑j=k−d(k)

δT (j) S2δ (j)

−

k−d(k)−1∑j=k−d

δT (j) S2δ (j) , (6)

where

ζ (k) =[Υ T (k) xT (k− τm) xT (k− d(k)) xT

(k− d

) ]T.

According to the definition of δ(j), for any matrices X , Y and Z thefollowing equations always hold:

2ζ T (k)X

[x(k)− x (k− τ)−

k−1∑j=k−τ

δ (j)

]= 0

2ζ T (k)Y

[x (k− τm)− x (k− d(k))−

k−τm−1∑j=k−d(k)

δ (j)

]= 0

2ζ T (k)Z

x (k− d(k))− x (k− d)− k−d(k)−1∑j=k−d

δ (j)

= 0. (7)

On the other hand, for any appropriately dimensioned matricesM ≥ 0 and N ≥ 0, the following identities hold:

612 X. Meng et al. / Automatica 46 (2010) 610–614

0 = τζ T (k)Mζ (k)−k−1∑j=k−τ

ζ T (k)Mζ (k),

0 =(d− τm

)ζ T (k)Nζ (k)−

k−τm−1∑j=k−d(k)

ζ T (k)Nζ (k)

−

k−d(k)−1∑j=k−d

ζ T (k)Nζ (k). (8)

Then, from (6)–(8), we have

1V (k) = 1V1(k)+1V2 (k)+1V3(k)+1V4(k)≤ ζ T (k)

(Ψ + Ψ T + Ξ T1 PΞ1 + Ξ

T2 Q2Ξ2

)ζ (k)

+ ζ T (k)(Φ +

(d− τm+ 1

)Ξ T2 RΞ2

)ζ (k)

+ ζ T (k)Ξ T3(τS1 +

(d− τm

)S2)Ξ3ζ (k)

−

k−1∑j=k−τ

ξ T (k, j)Π1ξ (k, j)−k−τm−1∑j=k−d(k)

ξ T (k, j)Π2ξ (k, j)

−

k−d(k)−1∑j=k−d

ξ T (k, j)Π3ξ (k, j) ,

where

ξ (k, j) =[ζ T (k) δT (j)

]T.

Using the Schur complement, (3) implies that Φ + Ψ + Ψ T +

Ξ T1 PΞ1 + ΞT2 (Q2 + (d − τm + 1)R)Ξ2 + Ξ

T3 τS1Ξ3 + Ξ

T3 (d −

τm)S2Ξ3 < 0, and Πi ≥ 0 (i = 1, 2, 3); that is, 1V (k) <−ε‖ζ (k)‖2, where ε is a positive scalar. Then, from the Lyapunovstability theory, we can conclude that the system (1) is asymptot-ically stable. �

Remark 3. The number of variables involved in the LMIs (3)–(5)is n2 [n(3m

2+ 18m + 41) + (3m + 11)]. However, the numbers

of variables in Hetel et al. (2008) and Xia et al. (2007) are n2 d(d +1)[(d+1)n+1] and n2 (d+1)[(d+1)n+1], respectively. For a fixedsystemwith state dimension n, the size of complexity in Hetel et al.(2008) and Xia et al. (2007) is related to d, while the complexity ofour result is only related to the number of partitionsm.

Remark 4. When estimating the upper bound on the differenceof a Lyapunov function, many previous methods use boundingtechniques which ignore some useful terms. However, retainingthose terms yields less conservative stability results.Mostmethodsemploy inequalities to estimate the cross products between twovectors, while the free-weighting matrices method circumventsthe utilization of bounding inequalities for cross products betweentwo vectors to reduce the conservatism. The advantage ofemploying free-weightingmatrices has been reported in a numberof earlier approaches; see He, Wu, Han, and She (2008), Parlacki(2007), and Parlakci (2008). Formore details, please refer to Xu andLam (2008).

Remark 5. The Lyapunov function (2) is different from theproposed Lyapunov function in Gao et al. (2004) in three ways: (A)V2(k) includes state xT (i−τ), xT (i−2τ) · · · xT (i−τm+2τ), xT (i−τm+τ) and information τ and d, which is not the case in Gao et al.(2004); (B) only V3(k) ,

∑−τm+1j=−d+1

∑k−1i=k−1+j x

T (i)Rx(i) is employed

to handle the time-varying delay while V2 and V3 are used in Gaoet al. (2004), which considerably complicates the proof; and (C)when calculating the forward difference of V4(k), we have not overbound this to −

∑k−1l=k−d η

T (m)Zη(m). Instead, we have separatedit into three parts, −

∑k−1j=k−τ δ

T (j)S1δ(j), −∑k−τm−1j=k−d(k) δ

T (j)S2δ(j)

and −∑k−d(k)−1j=k−d

δT (j)S2δ(j), and treated them individually usingdifferent free-weighting techniques.

Remark 6. Let m = 1, Q1 = Q2 = εI , where ε > 0 is a suff-iciently small scalar, Z = 0, S1 = S2, X = Y = [ XT 0n×3n ]T ,and M = N = diag{M, 03n} yields a form of Theorem 2 that isequivalent to Theorem 1 in Gao et al. (2004). That is, our theoremprovides more freedom in the selection ofm, Q1, Q2, and Z .

Remark 7. In the proof of Theorem2, d is separated into two parts:d = d(k) + d − d(k); and d − τm is separated into two parts:d− d = d(k)− d+ d− d(k). In contrast, as the inequalities(d−

k−1∑l=k−d(k)

)ζ T (k)MZ−11 M

T ζ (k) ≥ 0,

and((d− d

)−

k−d(k)−1∑l=k−d

)ζ T (k)SZ−11 S

T ζ (k) ≥ 0,

are employed in (6) in Gao and Chen (2007), d(k) and d− d(k) areincreased to d and d−d, respectively; that is, d = [d−d(k)]+d(k) ≤2d− d, which is clearly more conservative than our theorem.

The following proposition shows that conservatism is non-increasing as the integerm grows.

Proposition 8. Let d, h be the maximal delay obtained by Theorem 2for partitioning (m, τ ), (l, σ ) of a fixed lower bound, respectively. Ifm < l, then we have

d ≤ h.

Proof. For a chosen partition (τ ,m), due to the proof of Theorem2,one has(d− d+ 1

)xT (k)Rx (k)− xT (k− d(k)) Rx (k− d (k))

+1V1(k)+1V3(k)+1V4(k)+ (7)+ (8) < 0.

Replace (m, τ ) by (l, σ ), and Q1 by diag{Q1, εI(l−m)n}, X by[XT0n×(l−m)n]T , M by diag{M, 0(l−m)n} with matrices Y , Z beingreplaced by the same forms with X , and matrix N being replacedby the same forms withM , and Υ (k) by [xT (k)xT (k− σ) · · · xT (k−mσ +σ)xT (k−mσ) · · · xT (k− lσ +σ)]T , Υ (k− τ) by Υ (k−σ) =[xT (k−σ) · · · xT (k−mσ +σ)xT (k− d)xT (k−mσ) · · · xT (k− lσ +σ)]T , ζ (k) by [xT (k)xT (k− σ) · · · xT (k−mσ + σ)xT (k− d)xT (k−d(k))xT (k− d)xT (k−mσ) · · · xT (k− lσ + σ)]T , to get(d− d+ 1

)xT (k)Rx (k)− xT (k− d(k)) Rx (k− d (k))

+1V1(k)+1V3 (k)+1V4(k)+ ˜(7)+ ˜(8) < 0,

where X means the corresponding X in the above inequality for achosen partition (l, σ ). This inequality implies that for the partition(l, σ ) and h = d the system (1) is asymptotically stable. This alsoimplies that d ≤ h. �

Remark 9. The advantages of such augmentation over the aug-mentation [xT (i)xT (i− 1) · · · xT (i− τm)]T are twofold. On the onehand, the computational complexity is dependent on the partitionnumberm. For a systemwith dimension n, since the size of the de-cision variables increaseswith respect toO(m2), the computationalcomplexity will be increased as the partitioning becomes thinner.On the other hand, Proposition 8 proves that the conservatism isreduced as partitions grow. Thus our results provide the flexibilitythat allows us to trade off between complexity and performance ofthe stability analysis.

It is noted that in Theorem 2 we consider the case where thelower bound of the delay is assumed d ≥ 1. However, in applica-

X. Meng et al. / Automatica 46 (2010) 610–614 613

tions the lower bound of the delay could be zero, so we introducethe following proposition.

Proposition 10. Given a positive integer d, the system in (1) is asym-ptotically stable if there exist realmatrices P > 0,Q > 0, R > 0, S >0,M ≥ 0, X and Y satisfying[Φ + Ψ + Ψ T + dM Ξ

∗ −diag {P, R, S}

]< 0,[

M X∗ S

]≥ 0,

[M Y∗ S

]≥ 0,

where

Φ = −diag {P − Q , R,Q } ,

Ψ =[X Y

] [ I −I 0

0 I −I

],

Ξ =

[Ξ T1 P

√d+ 1Ξ T2 R

√dΞ T3 S

],

Ξ1 =[A B 0

], Ξ2 =

[I 0 0

], Ξ3 = Ξ1 − Ξ2.

Proof. Choose the following functional candidate:

V (k) , xT (k)Px (k)+k−1∑i=k−d

xT (i)Qx(i)

+

1∑j=−d+1

k−1∑i=k−1+j

xT (i) Rx(i)+−1∑i=−d

k−1∑j=k+i

δT (j) Sδ (j) , (9)

where δ(j) , x(j + 1) − x(j), and P > 0,Q > 0, R > 0, andS > 0. Then, by taking the forward difference of the (9) along thesolution of system (1), and applying the free-weightingmatrix ideain Theorem 2, we can conclude the above proposition. �

Remark 11. For a system with state dimension n and delay d(k)such that 0 ≤ d(k) ≤ d, the number of decision variables inProposition 10 is n2 (25n + 7), while the methods in Hetel et al.(2008) and Xia et al. (2007) require n2 d(d + 1)[(d + 1)n + 1] andn2 (d + 1)[(d + 1)n + 1], respectively. It can be deduced easilythat the computational burdens in Hetel et al. (2008) and Xia et al.(2007) are approximately d(d+1)2/25 and (d+1)2/25 higher thanthat involved in Proposition 10, respectively. Thus, the presentapproach is more favourable computationally especially when alarge time delay is involved.

Remark 12. The stability condition presented in this paper is forthe nominal system. However, it is not difficult to further extendthe results to uncertain systems, where the systemmatrices A andB in (1) contain parameter uncertainties either in norm-boundedor polytopic uncertain forms.

4. Illustrative examples

Example 13. Consider system (1) with

A =[0.8 00.05 0.9

], B =

[−0.1 0−0.2 −0.1

], (10)

which has been considered in Gao and Chen (2007) and Gao et al.(2004).Using the method proposed in Proposition 10, it is found that

system (10) is stable for 0 ≤ d(k) ≤ 17. To test the upper boundd = 17, the approach in Xia et al. (2007) needs 666 decisionvariables; however, the number of decision variables involved in

Table 1Allowable upper bound of d for various d.

Method d = 4 d = 12

Gao et al. (2004) 8 13Gao and Chen (2007) 13 16Theorem 2 17 (m = 2, τ = 2) 21 (m = 1, d = 12)Theorem 2 18 (m = 4, τ = 1) 22 (m = 2, τ = 6)

Table 2Allowable upper bound of dwhen d = 16.

m τ Number of variables d

1 16 138 242 8 195 244 4 345 258 2 789 25

Proposition 10 is only 57, which sufficiently demonstrates theefficiency of the proposed method. Now assume the lower boundis d = 2, and we are interested in the upper delay bound d belowwhich the above system is asymptotically stable for all d ≤ d(k) ≤d. It is obtained in Gao et al. (2004) that the maximum delay isd = 7. Using Gao and Chen (2007), one obtains d = 13. However,using the method proposed in Theorem 2, we obtain the upperdelay bound d = 17. A more detailed comparison is given inTable 1, where the achieved upper bounds of system (10) are listedfor their respective lower bounds. It can be seen that the methodproposed in this paper is significantly better than existing results.Table 2 shows the allowable upper bound for different (m, τ )whend = 16. From Table 2, we can see that the valuem becomes largerand the computation becomes heavierwith a steady increase in thenumber of decision variables. However, thismay not be necessarilyaccompanied with a prominent increase of d.

Example 14. We consider the system (1), where

A =[0.8 00 0.97

], B =

[−0.1 0−0.1 −0.1

].

Assuming that d(k) is time-varying and we apply Proposition 10.We obtain that asymptotic stability is guaranteed for all 0 ≤d(k) ≤ 15. Note that the condition of Lemma 2 in Fridman andShaked (2005) is only feasible for 0 ≤ d(k) ≤ 8. By the approachof Lemma 3 in Fridman and Shaked (2005), the stability intervals,starting fromnon-zero values, are [3, 10], [5, 11], [7, 12] and [9, 13].Moreover, using the approach in Hetel et al. (2008) shows that thesystem is asymptotically stable for 3 ≤ d(k) ≤ 13. We verify thatthe conditions of Theorem2are feasible for τ = 1, m = 1, d = 15,and thus the stability intervals are larger, [1, 15].

5. Conclusion

The problem of delay-dependent stability analysis for discrete-time systems with time-varying delay has been considered in thispaper. By introducing a new idea based on delay partitioning,stability criteria have been obtained by means of LMIs. Theresults reported in this paper are not only dependent on thedelay but also dependent on the partitioning, which aims atreducing the conservatism. It is also shown that the conservatismof the results is non-increasing when the number of partitions isincreased. Numerical examples are also presented to demonstratethe effectiveness and advantages of the proposed approach.

Acknowledgements

This work was supported by National Natural Science Founda-tion of China (60825303, 60834003), National Basic Research Pro-gram of China (2009CB320600), and RGC HKU 7031/07P.

614 X. Meng et al. / Automatica 46 (2010) 610–614

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