# Delay-Partitioning Approach to Stability of Linear Discrete-Time Systems with Interval-Like Time-Varying Delay

Authors: Priyanka Kokil; V. Krishna Rao Kandanvli; Haranath Kar
Journal: International Journal of Engineering Mathematics
 (2013)

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## Abstract

This paper is concerned with the problem of global asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state. By utilizing the concept of delay partitioning, a new linear-matrix-inequality-(LMI-) based criterion for the global asymptotic stability of such systems is proposed. The proposed criterion does not involve any free weighting matrices but depends on both the size of delay and partition size. The developed approach is extended to address the problem of global asymptotic stability of state-delayed discrete-time systems with norm-bounded uncertainties. The proposed results are compared with several existing results.


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## Body

## 1. Introduction

A source of instability for discrete-time systems is time delay which inevitably exists in various engineering systems. In many applications, time delays are unavoidable and must be taken into account in a realistic system design, for instance, chemical processes, echo cancellation, local loop equalization, multipath propagation in mobile communication, array signal processing, congestion analysis and control in high speed networks, neural networks, and long transmission line in pneumatic systems [1–6]. The stability analysis of time-delay systems has received considerable attention during the last two decades [3–7]. According to dependence of delay, the existing stability criteria are generally classified into two categories: delay-dependent criteria and delay-independent criteria. It is well known that delay-independent stability criteria are usually more conservative than the delay-dependent ones, especially if the size of time delay is small [6–10]. Therefore, much attention has been paid in recent years to the study of delay-dependent stability criteria.A number of publications relating to the delay-dependent stability of continuous time-delay systems have appeared (see, e.g., [10–17] and the references cited therein). In contrast, less attention has been paid to studying the problem of stability of discrete time-delay systems. Several delay-dependent criteria for the stability of discrete-time systems have appeared [3, 6, 7, 18, 19]. Reference [7] (see [6] also) presents a novel delay-dependent linear-matrix-inequality-(LMI-) based condition for the global asymptotic stability of linear discrete-time systems with interval-like time-varying delay. The criteria proposed in [6, 7] are less conservative with smaller numerical complexity than [19–23]. The delay partitioning approach has been efficiently applied in [24–30] to the stability analysis of systems with time-varying delays. In the context of stability analysis of linear discrete systems with time-varying delay, the delay partitioning concept is first utilized in [30]. The approaches in [29, 30] divide the lower bound of the time-varying delay into a number of partitions. The criteria in [29, 30] are not only delay dependent but also dependent on the partitioning size. Though the approaches in [29, 30] provide less conservative stability results than [19, 20], these approaches would lead to heavier computational burden and more complicated synthesis procedure.This paper studies the problem of stability analysis of linear discrete-time system with interval-like time-varying delay in the state. In this paper, inspired by the work of [6, 7, 30], an alternative to the approach presented in [29] for the stability analysis of linear discrete-time systems with interval-like time-varying delay in the state is brought out. The proposed method exploits the delay partitioning idea and does not introduce any free weighting matrices. The paper is organized as follows. Section 2 presents a description of the system under consideration. A novel LMI-based criterion for the global asymptotic stability of discrete-time state-delayed systems is proposed in Section 3. The proposed criterion depends on the size of delay as well as partition size. In Section 4, the approach is extended to derive global asymptotic stable conditions for delayed discrete-time systems with norm-bounded uncertainties. Finally, in Section 5, numerical examples are given to illustrate the effectiveness of the presented results.


## 2. System Description

The following notations are used throughout the paper:Rp×q: Set of p×q real matrices,Rp: Set of p×1 real vectors,I: Identity matrix of appropriate dimension; the order is specified in subscript as the need arises,0: Null matrix or null vector of appropriate dimension; the orders are specified in subscripts as the need arises,BT: Transpose of the matrix (or vector) B,B>0:B is positive-definite symmetric matrix,B<0:B is negative-definite symmetric matrix.The system under consideration is given by(1)x(k+1)=Ax(k)+A1x(k-d(k)),(2)x(k)=ϕ(k),k=-h2,-h2+1,…,0,
where x(k)∈Rn is the system state vector, A and A1 are constant matrices with appropriate dimensions, d(k) is a positive integer representing interval-like time-varying delay satisfying
(3)1≤h1≤d(k)≤h2,
where h1 and h2 are known positive integers representing the lower and upper delay bounds, respectively, and ϕ(k) is an initial value at time k. Let the lower bound of the delay h1 be divided into mnumber of partitions such that
(4)h1=τm,
where τ is an integer representing partition size.


## 3. Proposed Criterion

In this section, inspired by the work of [6, 7, 30], an LMI-based criterion for the global asymptotic stability of system (1)–(4) is established.The main result may be stated as follows.Theorem 1.
For given positive integersτ, m, and h2, the system in (1)–(4) is asymptotically stable if there exist real matrices P=PT>0, Qi=QiT>0(i=1,2,3), and Zi=ZiT>0(i=1,2) such that
(5)Ψ1=Ψ-[0n×mnIn-In0n]TZ2×[0n×mnIn-In0n]<0,Ψ2=Ψ-[0n×(m+1)nIn-In]TZ2×[0n×(m+1)nIn-In]<0,
where
(6)Ψ=Λ1TΦ1Λ1+Λ2TΦ2Λ2+Λ3TΦ3Λ3-Λ4TΦ4Λ4+Λ1TΦ5Λ3+Λ3TΦ5TΛ1-Λ5TZ1Λ5+Λ1TZ1Λ5+Λ5TZ1Λ1-Λ6TZ2Λ6+Λ3TZ2Λ4+Λ4TZ2Λ3+Λ6TZ2Λ3+Λ3TZ2Λ6,(7)Λ1=[In0n×(m+2)n],Λ2=[Imn0mn×3n0mn×nImn0mn×2n],(8)Λ3=[0n×(m+1)nIn0n],Λ4=[0n×(m+2)nIn],(9)Λ5=[0nIn0n×(m+1)n],Λ6=[0n×mnIn0n×2n],(10)Φ1=ATPA-P+Q2+(h2-τm+1)Q3-Z1+(A-In)T(τ2Z1+(h2-τm)2Z2)(A-In),(11)Φ2=[Q100-Q1],(12)Φ3=A1T(P+τ2Z1+(h2-τm)2Z2)A1-Q3-2Z2,(13)Φ4=Q2+Z2,(14)Φ5=ATPA1+(A-In)T(τ2Z1+(h2-τm)2Z2)A1.Proof.
Consider a Lyapunov function [29, 30]
(15)V(k)=xT(k)Px(k)+∑i=k-τk-1ΥT(i)Q1Υ(i)+∑i=k-h2k-1xT(i)Q2x(i)+∑j=-h2-τm∑i=k+jk-1xT(i)Q3x(i)+∑j=-τ-1∑i=k+jk-1τΔxT(i)Z1Δx(i)+∑j=-h2-τm-1∑i=k+jk-1(h2-τm)ΔxT(i)Z2Δx(i),
where
(16)Υ(i)=[xT(i)xT(i-τ)⋯xT(i-(m-1)τ)]T,(17)Δx(i)=x(i+1)-x(i).
Taking the forward difference of (15) along the solution of the system (1)-(2), we have
(18)ΔV(k)=V(k+1)-V(k)=[Ax(k)+A1x(k-d(k))]TP[Ax(k)+A1x(k-d(k))]=-xT(k)Px(k)+ΥT(k)Q1Υ(k)-ΥT(k-τ)Q1Υ(k-τ)=+xT(k)Q2x(k)-xT(k-h2)Q2x(k-h2)=+(h2-τm+1)xT(k)Q3x(k)+ΔxT(k)τ2Z1Δx(k)=+ΔxT(k)(h2-τm)2Z2Δx(k)-∑i=k-h2k-τmxT(i)Q3x(i)=-∑i=k-τk-1τΔxT(i)Z1Δx(i)=-∑i=k-h2k-τm-1(h2-τm)ΔxT(i)Z2Δx(i).=
Using Lemma 1 in [22], we obtain [6, 7]
(19)-∑i=k-τk-1τΔxT(i)Z1Δx(i)≤-(x(k)-x(k-τ))TZ1(x(k)-x(k-τ)).
Note that
(20)-∑i=k-h2k-τmxT(i)Q3x(i)≤-xT(k-d(k))Q3x(k-d(k)).
Define
(21)ζ(k)=[ΥT(k)xT(k-τm)xT(k-d(k))xT(k-h2)]T.
It follows from (1) and (17) that
(22)Δx(k)=(A-In)x(k)+A1x(k-d(k)).
Now, we have the following relation [6, 7]:
(23)-∑i=k-h2k-τm-1(h2-τm)ΔxT(i)Z2Δx(i)=-∑i=k-h2k-d(k)-1(h2-τm)ΔxT(i)Z2Δx(i)-∑i=k-d(k)k-τm-1(h2-τm)ΔxT(i)Z2Δx(i)=-∑i=k-h2k-d(k)-1(h2-d(k))ΔxT(i)Z2Δx(i)-∑i=k-h2k-d(k)-1(d(k)-τm)ΔxT(i)Z2Δx(i)-∑i=k-d(k)k-τm-1(h2-d(k))ΔxT(i)Z2Δx(i)-∑i=k-d(k)k-τm-1(d(k)-τm)ΔxT(i)Z2Δx(i).
Letting β=(d(k)-τm)/(h2-τm), one obtains [6]
(24)-∑i=k-h2k-d(k)-1(d(k)-τm)ΔxT(i)Z2Δx(i)=-β∑i=k-h2k-d(k)-1(h2-τm)ΔxT(i)Z2Δx(i)≤-β∑i=k-h2k-d(k)-1(h2-d(k))ΔxT(i)Z2Δx(i),-∑i=k-d(k)k-τm-1(h2-d(k))ΔxT(i)Z2Δx(i)=-(1-β)∑i=k-d(k)k-τm-1(h2-τm)ΔxT(i)Z2Δx(i)≤-(1-β)∑i=k-d(k)k-τm-1(d(k)-τm)ΔxT(i)Z2Δx(i).
Applying Lemma 1 in [22] and using (23)-(24), we have [6, 7]
(25)-∑i=k-h2k-τm-1(h2-τm)ΔxT(i)Z2Δx(i)≤-(x(k-d(k))-x(k-h2))T≤×Z2(x(k-d(k))-x(k-h2))≤-(x(k-τm)-x(k-d(k)))T≤×Z2(x(k-τm)-x(k-d(k)))≤-β(x(k-d(k))-x(k-h2))T≤×Z2(x(k-d(k))-x(k-h2))≤-(1-β)(x(k-τm)-x(k-d(k)))T≤×Z2(x(k-τm)-x(k-d(k))).
Employing (18)–(22) and (25), we have the following inequality:(26)ΔV(k)≤ζT(k)[(1-β)Ψ1+βΨ2]ζ(k).
Since 0≤β≤1, ΔV(k)<0 for all nonzero ζ(k) if  (5) hold true. This completes the proof of Theorem 1.Remark 2.
To prove Theorem1, a methodology similar to [6, 7] has been adopted.Remark 3.
It may be noted that the inequalities (5) are LMIs and can be effectively solved by using MATLAB LMI toolbox [31, 32].Remark 4.
For a givenτ and m, the allowable maximum value of h2 for guaranteeing the global asymptotic stability of system (1)–(4) can be obtained by iteratively solving (5).Remark 5.
Form=1, after some algebraic manipulations, it can be shown that Theorem 1 is equivalent to Theorem 1 in [7]. The parameters m and τ (subject to (4)) in Theorem 1 represent some additional degrees of freedom, that is, in comparison with Theorem 1 in [7] which is free of these parameters.Remark 6.
A comparison of the number of the decision variables involved in several recent stability results is summarized in Table1. It may be observed that the size of complexity in [6, 7, 18, 19, 23] is only related to state dimension n, whereas the complexity of  [29, Theorem 3.1], [30, Theorem 2], and Theorem 1 depends on both n and m.Table 1
Comparison of the number of decision variables involved in various methods.
Methods
Number of the decision variables
Theorem1 in [19]
(
23
n
2
+
5
n
)
/
2
Theorem3 in [19]
(
67
n
2
+
9
n
)
/
2
Theorem1 in [23]
9
n
2
+
3
n
Theorem1 in [18]
13
n
2
+
5
n
Proposition2 in [6]
3
n
2
+
3
n
Theorem1 in [7]
3
n
2
+
3
n
Theorem2 in [30]
(n/2)[n(3m2+18m+41)+(3m+11)]
Theorem3.1 in [29]
(
n
/
2
)
[
n
(
m
2
+
4
m
+
19
)
+
(
m
+
5
)
]
Theorem1
(
n
/
2
)
[
n
(
m
2
+
5
)
+
(
m
+
5
)
]The total number of scalar decision variables of Theorem1 is D1=(n/2)[n(m2+5)+(m+5)], and the total row size of the LMIs is L1=n(3m+11). The numerical complexity of Theorem 1 is proportional to L1D13 [14]. The total number of scalar decision variables of Theorem 2 in [30] is D2=(n/2)[n(3m2+18m+41)+(3m+11)], the total row size of the LMIs is L2=n(7m+31), and the numerical complexity is proportional to L2D23. Therefore, Theorem 1 has much smaller numerical complexity than Theorem 2 in [30]. Since no free weighting matrix has been introduced in the presented method, Theorem 1 involves less number of decision variables as compared to Theorem 3.1 in [29] (see Table 1). Thus, in general, Theorem 1 is numerically less complex than Theorem 3.1 in [29].Remark 7.
Using similar steps as in the proof of Proposition 8 in [30], it is easy to establish that the conservatism of the stability result obtained via Theorem 1 is nonincreasing as the number of partitions increases.


## 4. Extensions to Uncertain State-Delayed Discrete-Time Systems

In this section, we extend the previously discussed approach to derive global asymptotic stable conditions for delayed discrete-time systems with norm-bounded uncertainties.Consider the system given by(27)x(k+1)=(A+ΔA)x(k)+(A1+ΔA1)x(k-d(k)).
The parameter uncertainties ΔA and ΔA1 are assumed to be norm-bounded and of the following form:
(28)[ΔAΔA1]=HF[E0E1],
where H∈Rn×q and Ei∈Rp×n(i=0,1) are known constant matrices and F∈Rq×p is an unknown matrix which satisfies
(29)FTF≤I.
The parameter uncertainties ΔA and ΔA1 are said be admissible if both (28) and (29) are satisfied.The following lemma is needed in the proof of our next result.Lemma 8 (see [5, 25]).
LetΣ,Γ,F, and M be the real matrices of appropriate dimensions with M satisfying M=MT; then
(30)M+ΣFΓ+ΓTFTΣT<0,
for all FTF≤I, if and only if there exists a scalar ε>0 such that
(31)M+ε-1ΣΣT+εΓTΓ<0.Theorem 9.
For given scalarsh1 and h2, system described by (27)–(29) and (2)–(4) is globally asymptotically stable for all the admissible uncertainties, if there exist real matrices P=PT>0, Qi=QiT>0(i=1,2,3), and  Zi=ZiT>0(i=1,2) and a scalar ε>0 such that the following LMIs hold:
(32)[Ψ-1+εE-TE-Δ1TPΔ2TZ-0(m+3)n×qPΔ1-P0nPHZ-Δ20n-Z-Z-H0q×(m+3)nHTPHTZ--εIq]<0,(33)[Ψ-2+εE-TE-Δ1TPΔ2TZ-0(m+3)n×qPΔ1-P0nPHZ-Δ20n-Z-Z-H0q×(m+3)nHTPHTZ--εIq]<0,
where(34)Ψ-1=Ψ~-Λ7TZ2Λ7,Ψ-2=Ψ~-Λ8TZ2Λ8,(35)Ψ~=Λ1TΦ~1Λ1+Λ2TΦ~2Λ2+Λ3TΦ~3Λ3-Λ4TΦ~4Λ4-Λ5TZ1Λ5+Λ1TZ1Λ5+Λ5TZ1Λ1-Λ6TZ2Λ6+Λ3TZ2Λ4+Λ4TZ2Λ3+Λ6TZ2Λ3+Λ3TZ2Λ6,(36)Λ7=[0n×mnIn-In0n],Λ8=[0n×(m+1)nIn-In],Φ~1=-P+Q2+(h2-τm+1)Q3-Z1,(37)Φ~2=[Q100-Q1],(38)Φ~3=-Q3-2Z2,Φ~4=Q2+Z2,(39)Z-=τ2Z1+(h2-τm)Z2,Δ1=[A0n×mnA10n],(40)Δ2=[A-In0n×mnA10n],E-=[E00p×mnE10p×n],
and Λi(i=1,…,6) in (35) are given by (7)–(9).Proof.
Applying Theorem1, the sufficient conditions for the global asymptotic stability of system described by (27)–(29) and (2)–(4) are obtained as follows:
(41)[ATP(A-In)TZ-Ψ-10mn×n0mn×nA1TPA1TZ-0n0nPA0n×mnPA10n-P0nZ-(A-In)0n×mnZ-A10n0n-Z-]+[0(m+3)n×qPHZ-H]F[E00p×mnE10p×3n]+[E0T0mn×pE1T03n×p]FT[0q×(m+3)nHTPHTZ-]<0,(42)[ATP(A-In)TZ-Ψ-20mn×n0mn×nA1TPA1TZ-0n0nPA0n×mnPA10n-P0nZ-(A-In)0n×mnZ-A10n0n-Z-]+[0(m+3)n×qPHZ-H]F[E00p×mnE10p×3n]+[E0T0mn×pE1T03n×p]FT[0q×(m+3)nHTPHTZ-]<0.
By Lemma 8 and Schur complement, it can be shown that (41) and (42) are equivalent to (32) and (33), respectively. This completes the proof of Theorem 9.Remark 10.
Withm=1, Theorem 9 reduces to an equivalent form of Theorem 4 in [7]. The parameters m and τ (subject to (4)) in Theorem 9 may be treated as additional degrees of freedom, that is, in comparison with Theorem 4 in [7] which is free of such parameters.


## 5. Numerical Examples

To demonstrate the applicability of the presented results and compare them with previous results, we now consider the following examples.Example 1 (see [6, 7, 19, 20, 29]).
Consider the delayed discrete-time system (1) with the following parameters:
(43)A=[0.800.050.9],A1=[-0.10-0.2-0.1].For givenτ and m, the admissible maximum value of h2 guaranteeing the asymptotically stability of the delayed discrete-time system can be obtained by solving the LMIs (5). For a given h1, a comparison of the maximum values of  h2 calculated by different methods is summarized in Table 2, from which one can easily see that the stability conditions obtained in this paper are less conservative than those in [7, 19, 20].Table 2
Comparison of the maximum values ofh2 for given h1.
Methods
h
1
=
4
h
1
=
6
h
1
=
10
h
1
=
12
h
1
=
20
Theorem1 in [20]
8
9
12
13
20
Theorem1 in [19]
13
14
15
16
22
Proposition2 in [6]
15
16
18
19
25
Theorem1 in [7]
15
16
18
19
25
Theorem1
15(m=1,τ=4)
16(m=1,τ=6)
18(m=1,τ=10)
19(m=1,τ=12)
25(m=1,τ=20)
15(m=2,τ=2)
16(m=2,τ=3)
18(m=2,τ=5)
20(m=2,τ=6)
26(m=2,τ=10)A comparison of [29, Table 2] and Table 2 reveals that Theorem 3.1 in [29] may provide a larger upper delay bound h2 than Theorem 1. However, as discussed in Remark 6, Theorem 1 involves less number of decision variables as compared to Theorem 3.1 in [29].Example 2 (see [7, 19]).
Consider the following delayed discrete-time system with parameter uncertainty:(44)x(k+1)=[0.8+α(k)000.9]x(k)+[-0.10-0.1-0.1]x(k-d(k)),
where x(k)=[x1(k)x2(k)]T and |α(k)|≤α-.  System (44) can be represented in the form of (27) with
(45)A=[0.8000.9],A1=[-0.10-0.1-0.1],H=[α-0],E0=[01],E1=[00],F=α(k)α-.
For given h1 and h2, we wish to find α- such that the uncertain system (44) is asymptotically stable for any |α(k)|≤α-. Table 3 presents a comparison of the values of  α- obtained by various methods. From Table 3, it is clear that Theorem 9 can provide less conservative results than [7, 19, 20]. In the situation where 20≤d(k)≤26, the stability criteria in [7, 19, 20] are invalid whereas the value of α- obtained by Theorem 9 is 0.0837 for m=5,τ=4.Table 3
Comparison of upper boundα- for given  h1 and h2.
Methods
2
≤
d
(
k
)
≤
7
3
≤
d
(
k
)
≤
9
5
≤
d
(
k
)
≤
10
6
≤
d
(
k
)
≤
12
10
≤
d
(
k
)
≤
15
20
≤
d
(
k
)
≤
26
Theorem5 in [20]
0.0830
Infeasible
Infeasible
Infeasible
Infeasible
Infeasible
Corollary2 in [19]
0.1901
0.1457
0.1313
0.0906
0.0655
Infeasible
Theorem4 in [7]
0.1920
0.1548
0.1425
0.1146
0.1023
Infeasible
Theorem9
0.1927(m=2,τ=1)
0.1560(m=3,τ=1)
0.1476(m=5,τ=1)
0.1194(m=2,τ=3)
0.1080(m=2,τ=5)
0.0834(m=2,τ=10) 0.0837(m=5,τ=4)


## 6. Conclusion

In this paper, we have considered the problem of global asymptotic stability of linear discrete-time systems with interval-like time-varying delay in the state. By utilizing the concept of delay partitioning, an LMI-based criterion for the global asymptotic stability of such systems has been established. The proposed criterion depends on both the size of delay and partition size. With  the help of numerical examples, it has been illustrated that the proposed stability condition may provide less conservative result than most of the existing results [6, 7, 19, 20] due to the delay partitioning, and it may become even less conservative when the partitioning goes finer. Since no free weighting matrix has been introduced in the presented method, Theorem 1 involves less number of decision variables as compared to Theorem 3.1 in [29]. However, a comparison of [29, Table 2] and Table 2 reveals that Theorem 3.1 in [29] may provide a larger upper delay bound h2 than Theorem 1. The proposed criterion is numerically less complex than [30]. The method has been extended to deal with the problem of global asymptotic stability of state-delayed discrete-time systems with norm-bounded uncertainties.



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