# Efficient Approach to Stability Analysis of Discrete-Time Systems with Time-Varying Delay

Authors: Sung Wook Yun; Sung Hyun Kim
Journal: Mathematical Problems in Engineering
 (2015)

---

## Abstract

This paper aims at deriving an efficient criterion for the robust stability analysis of discrete-time systems
with time-varying delay. In the derivation, to obtain a larger stability region under the requirement
of less computational complexity, this paper proposes a valuable method capable of establishing a less
conservative stability criterion without using the free-weighting approach and an extremely augmented
state. In parallel, the stabilization problem of systems with time-delayed control input is addressed in
connection with the derived stability criterion.


---

## Body

## 1. Introduction

Over the last few decades, research on stability analysis of time-delay systems has rapidly accelerated owing to two main reasons. One is that such systems offer suitable mathematical models that can represent practical engineering systems with finite but uncertain signal propagation delays, such as biological systems, network systems, and nuclear reactors [1–3]. The other is that time-delay can often act as a critical factor that leads to performance degradation and instability of the systems under consideration [4–6]. Meanwhile, with the growing interest, significant progress has been made toward enlarging the feasible region of a stability criterion (referred to here as “stability region”) and reducing its computational complexity. In other words, numerous investigations and research have been carried out to establish further improved stability criteria for time-delay systems by taking one of the following approaches with the use of a more attractive Lyapunov-Krasovskii functional [7–11]: the free-weighting matrix approach [11–13], the descriptor system approach [14–16], the Jensen inequality approach [17–19], or the delay-partitioning approach [20, 21].Recently, the use of the Jensen inequality approach has attracted great attention from the control community since it requires fewer decision variables than other approaches (see [18, 22, 23] and references therein). In addition, the appearance of the reciprocally convex technique [22] has promoted the use of such an approach as a way to reduce the computational complexity and the conservatism of stability criteria. However, as reported in [24], most of the results of these studies have been confronted with a challenge to establish less conservative stability criteria in terms of performance behavior. Thus, [24] has fully exploited the zero equality terms, introduced by [25], in the process of deriving stability criteria. After that, [10, 11] have widely extended the stability criteria in accordance with the structure of Lyapunov-Krasovskii functional containing triple summation terms. However, since the use of such an augmented Lyapunov-Krasovskii function with more terms poses significant computational burdens, there is a need to explore a useful method capable of reducing the computational complexity as well as improving the performance of stability criteria.Motivated by the above concerns, this paper is focused on deriving an efficient criterion for the robust stability analysis of discrete-time systems with time-varying delay. To be specific, the attention of this paper is paid for simultaneously achieving the following two goals: deceasing the computational complexity and improving the performance of a stability criterion. To accomplish such efficiency, this paper proposes a valuable method capable of deriving a less conservative stability criterion without using the free-weighting approach and an extremely augmented state, which plays an important role in reducing the computational complexity caused by [10, 11]. As a result, under the requirement of much less computational complexity, the stability region in the present study is enlarged to the same size as those of [10, 11]. Moreover, the stabilization problem of systems with time-delay control input is addressed in connection with the derived stability criterion. Finally, three numerical examples are provided to illustrate the efficiency of the proposed stability and stabilization criteria.This paper is organized as follows. Section2 provides a system description and useful properties. Section 3 introduces a Lyapunov-Krasovskii functional for deriving the stability criterion of the time-delay system. Section 4 presents the state-feedback controller for the system with delayed control input. Section 5 shows simulation results for validating the proposed results. Finally, Section 6 presents the conclusion along with a summary.Notation. The notations X≥Y and X>Y mean that X-Y is positive semidefinite and positive definite, respectively. In symmetric block matrices, (∗) is used as an ellipsis for terms that are induced by symmetry. For any square matrix Q, He[Q]=Q+QT. For any discrete-time function gk, Δ[gk] denotes its forward difference as Δ[gk]=gk+1-gk.


## 2. System Description and Useful Properties

Let us consider the following time-delay system:(1)xk+1=Axk+Adxk-dk,where xk∈Rnx and xk-d(k)∈Rnx are the state and the delayed state, respectively. Here, the state delay d(k) is assumed to be of an interval time-varying type integer: d_≤d(k)≤d¯, where d_ and d¯ are known positive integers. To facilitate the derivation of the main result, we set d0=0, d1=d_, d2=d(k), and d3=d¯ and use the following notations:(2)Φpq1ηi=∑i=k-dqk-dp-1ηi,Φpq2ηi=∑j=dp+1dq∑i=k-jk-1ηi,where p∈{0,1,2}, q∈{1,2,3∣q>p}, and ηi denotes any scalar or vector-valued function.Property 1.
For(p,q)∈{(0,1),(0,3),(1,3)}, the following properties hold:(3)ΔΦpq1ηi=Φpq1Δηi=ηk-dp-ηk-dq,(4)ΔΦpq2ηi=Φpq2Δηi=Φpq11ηk-Φpq1ηi,where Φpq1(1)=dq-dp.Property 2.
Let us consider two time-varying parameters of the following form:(5)θ1=Φ1211Φ1311≥0,θ2=Φ2311Φ1311≥0.Then, in the sense that Φ121(1)+Φ231(1)=Φ131(1), it follows that θ1+θ2=1.Lemma 1 (see [10, 17]).
For any vector-valued functionχi and positive-definite matrix Q, the following inequalities hold:(6)-Φpq1χiTQχi≤-1Φpq11Φpq1χiTQΦpq1χi.


## 3. Stability Analysis

LetΔxi=xi+1-xi and choose a Lyapunov-Krasovskii functional of the following form:(7)V1,k=xkΦ011xiΦ131xiTP1P2P3∗P4P5∗∗P6≜PxkΦ011xiΦ131xi,V2,k=Φ011xiTQ1xi+Φ031xiTQ2xi,V3,k=Φ0111·Φ012xiΔxiTR1100R12≜R1xiΔxi,V4,k=Φ1311·Φ132xiΔxiTR2100R22≜R2xiΔxi,where P, Q1, Q2, R1, and R2 are taken to be positive definite. To facilitate later steps, we define an augmented state ζk as(8)ζk=xkTxk-d1Txk-d2Txk-d3TΔxkTΦ011xiTΦ121xiTΦ231xiTT∈Rnζ,nζ=8nxand establish block entry matrices ei such that xk=e0ζk, xk-d1=e1ζk, xk-d2=e2ζk, xk-d3=e3ζk, Δxk=e4ζk, Φ011(xi)=e5ζk, Φ121(xi)=e6ζk, and Φ231(xi)=e7ζk.Property 3 (see [25]).
For symmetric matricesX0, X1, and X2, the following equalities hold:(9)0=Φ0111·ζkTe0TX0e0-e1TX0e1ζk-Φ011ΔxiX0Δxi+2xi,0=Φ1311·ζkTe1TX1e1-e2TX1e2ζk-Φ121ΔxiX1Δxi+2xi,0=Φ1311·ζkTe2TX2e2-e3TX2e3ζk-Φ231ΔxiX2Δxi+2xi.
The following theorem presents the delay- and range-dependent stability criterion for (1).Theorem 2.
Letd1=d_ and d3=d¯ be prescribed and define δ=d3-d1. System (1) is asymptotically stable for any time-varying d(k) satisfying d(k)∈[d_,d¯], if there exist matrices {Si}i=1,…,4∈Rnx×nx, {Zi}i=0,4∈Rnx×nx and symmetric matrices P∈R3nx×3nx, {Qi}i=1,2∈Rnx×nx, {R1i}i=1,2∈Rnx×nx, {R2i}i=1,2∈Rnx×nx, and {Xi}i=0,1,2∈Rnx×nx such that(10)
0
>
Ψ
1
+
Ψ
2
+
Ψ
3
+
Ψ
4
+
Ψ
5
,
(11)
0
≤
R
21
X
1
S
1
S
2
∗
R
22
+
X
1
S
3
S
4
∗
∗
R
21
X
2
∗
∗
∗
R
22
+
X
2
,where {Ψi}i=1,…,5 are defined in (13), (15), (18), (20), and (22), respectively.Proof.
The forward difference ofV1,k becomes(12)
Δ
V
1
,
k
=
Δ
x
k
+
x
k
Φ
01
1
Δ
x
i
+
x
i
Φ
13
1
Δ
x
i
+
x
i
T
P
Δ
x
k
+
x
k
Φ
01
1
Δ
x
i
+
x
i
Φ
13
1
Δ
x
i
+
x
i
-
x
k
Φ
01
1
x
i
Φ
13
1
x
i
T
P
x
k
Φ
01
1
x
i
Φ
13
1
x
i
=
ζ
k
T
e
4
e
0
e
0
-
e
1
e
5
e
1
-
e
3
e
6
+
e
7
T
P
1
P
1
P
2
P
2
P
3
P
3
P
1
0
P
2
0
P
3
0
P
2
T
P
2
T
P
4
P
4
P
5
P
5
P
2
T
0
P
4
0
P
5
0
P
3
T
P
3
T
P
5
T
P
5
T
P
6
P
6
P
3
T
0
P
5
T
0
P
6
0
e
4
e
0
e
0
-
e
1
e
5
e
1
-
e
3
e
6
+
e
7
ζ
k
,
from
3
=
ζ
k
Ψ
1
ζ
k
,where(13)Ψ1=e4TP1e4+Hee4TP1e0+e4TP2e0-e1+e4TP2e5+e4TP3e1-e3+e4TP3e6+e7+Hee0TP2e0-e1+e0TP3e1-e3+e0-e1TP4e0-e1+Hee0-e1TP4e5+e0-e1TP5e1-e3+e0-e1TP5e6+e7+Hee5TP5e1-e3+e1-e3TP6e1-e3+Hee1-e3TP6e6+e7.Letting x¯iT=xiTΔxiT, the forward differences of V2,k, V3,k, and V4,k are, respectively, given by(14)ΔV2,k=xkTQ1xk-xk-d1TQ1xk-d1+xkTQ2xk-xk-d3Q2xk-d3=ζkTΨ2ζkfrom3,ΔV3,k=Φ01112·x¯kTR1x¯k-Φ0111·Φ011x¯iR1x¯ifrom4,ΔV4,k=Φ13112·x¯kTR2x¯k-Φ1311·Φ131x¯iTR2x¯ifrom4,where(15)Ψ2=e0TQ1e0-e1TQ1e1+e0TQ2e0-e3TQ2e3.Here, with the help of (9), ΔV3,k and ΔV4,k can be converted into(16)ΔV3,k=Φ01112·x¯kTR1x¯k-Φ0111·Φ011x¯iTR1x¯i+Φ0111·ζkTe0TX0e0-e1TX0e1ζk-Φ0111·Φ011ΔxiX0Δxi+2xi,ΔV4,k=Φ13112·x¯kTR2x¯k-Φ1311·Φ131x¯iTR2x¯i+Φ1311·ζkTe1TX1e1-e2TX1e2ζk-Φ1311·Φ121ΔxiX1Δxi+2xi+Φ1311·ζkTe2TX2e2-e3TX2e3ζk-Φ1311·Φ231ΔxiX2Δxi+2xi.In particular, by Lemma 1, the following inequality holds:(17)-Φ0111·Φ011x¯iTR1x¯i-Φ0111·Φ011ΔxiX0Δxi+2xi=-Φ0111·Φ011x¯iTR11X0∗R12+X0≜X0x¯i≤-Φ011x¯iTX0Φ011x¯i=-ζkTe5e0-e1TR11X0∗R12+X0e5e0-e1ζk,which implies ΔV3,k≤ζkTΨ3ζk, where(18)Ψ3=d12e0TR11e0+e4TR12e4+d1e0TX0e0-d1e1TX0e1-e5TR11e5-Hee5TX0e0-e1-e0-e1TR12+X0e0-e1.Likewise, we can obtain(19)
-
Φ
13
1
1
·
Φ
13
1
x
¯
i
T
R
2
x
¯
i
-
Φ
13
1
1
·
Φ
12
1
Δ
x
i
X
1
Δ
x
i
+
2
x
i
-
Φ
13
1
1
·
Φ
23
1
Δ
x
i
X
2
Δ
x
i
+
2
x
i
=
-
Φ
13
1
1
·
Φ
12
1
x
¯
i
T
R
21
X
1
∗
R
22
+
X
1
≜
X
1
x
¯
i
-
Φ
13
1
1
·
Φ
23
1
x
¯
i
T
R
21
X
2
∗
R
22
+
X
2
≜
X
2
x
¯
i
≤
-
Φ
13
1
1
Φ
12
1
1
·
Φ
12
1
x
¯
i
T
X
1
Φ
12
1
x
¯
i
-
Φ
13
1
1
Φ
23
1
1
·
Φ
23
1
x
¯
i
T
X
2
Φ
23
1
x
¯
i
=
-
1
θ
1
Φ
12
1
x
¯
i
T
X
1
Φ
12
1
x
¯
i
-
1
θ
2
Φ
23
1
x
¯
i
T
X
2
Φ
23
1
x
¯
i
from  Property
2
=
-
Φ
12
1
x
¯
i
Φ
23
1
x
¯
i
T
X
1
S
∗
X
2
Φ
12
1
x
¯
i
Φ
23
1
x
¯
i
-
θ
2
θ
1
Φ
12
1
x
¯
i
-
θ
1
θ
2
Φ
23
1
x
¯
i
T
X
1
S
∗
X
2
θ
2
θ
1
Φ
12
1
x
¯
i
-
θ
1
θ
2
Φ
23
1
x
¯
i
≤
-
e
6
e
1
-
e
2
e
7
e
2
-
e
3
T
R
21
X
1
∗
R
22
+
X
1
S
1
S
2
S
3
S
4
≜
S
∗
R
21
X
2
∗
R
22
+
X
2
e
6
e
1
-
e
2
e
7
e
2
-
e
3
under
11
,which implies ΔV4,k≤ζkTΨ4ζk, where(20)Ψ4=δ2e0TR21e0+e4TR22e4+δe1TX1e1-δe2TX1e2+δe2TX2e2-δe3TX2e3-e6TR21e6-Hee6TX1e1-e2+e6TS1e7+e6TS2e2-e3-e1-e2TR22+X1e1-e2-Hee1-e2TS3e7+e1-e2TS4e2-e3-e7TR21e7-Hee7TX2e2-e3-e2-e3TR22+X2e2-e3.Hence, the forward difference of Vk satisfies(21)ΔVk≤ζkTΨ1+Ψ2+Ψ3+Ψ4ζk.Moreover, by adding 0=ζkTΨ5ζk to (21), where(22)Ψ5=Hee0TZ0+e4TZ4e4+I-Ae0-Ade2,we can obtain ΔVk≤ζkT(Ψ1+Ψ2+Ψ3+Ψ4+Ψ5)ζk. Therefore, the stability criterion is given by (10) and (11).Remark 3.
For the givennx, the number of scalar variables (NSVs) used in Theorem 2 is given as 15nx2+6nx. Our approach leads to a significant decrease in the computational burden compared with [10], [11], and [24] that demand 27nx2+9nx, 90.5nx2+14.5nx, and 16nx2+9nx scalar variables, respectively.Remark 4.
To make up for the weakness in [10, 11, 24], this paper proposes a valuable method capable of deriving a less conservative stability criterion without using the free-weighting approach and the inclusion of Δxk-d1, Δxk-d2, and Δxk-d3 in the augmented state.Remark 5.
From (10), it follows that 0>e4(Ψ1+Ψ2+Ψ3+Ψ4+Ψ5)e4T=P1+R12+R22+He[Z4]. As a result, Z4+Z4T<0; that is, Z4 is nonsingular since P1, R12, and R22 are positive definite.As a by-product of Theorem2, we can obtain a robust stability criterion for(23)xk+1=Axk+Adxk-dk+Gpk,qk=Exk+Edxk-dk,where pk∈Rnp and qk∈Rnq such that pk=Δkqk and ΔkTΔk≤I.Corollary 6.
Letd1=d_ and d3=d¯ be prescribed and define δ=d3-d1. System (1) is asymptotically stable for any time-varying d(k) satisfying d(k)∈[d_,d¯], if there exist matrices {Si}i=1,…,4∈Rnx×nx, {Zi}i=0,4∈Rnx×nx and symmetric matrices P∈R3nx×3nx, {Qi}i=1,2∈Rnx×nx, {R1i}i=1,2∈Rnx×nx, {R2i}i=1,2∈Rnx×nx, and {Xi}i=0,1,2∈Rnx×nx such that (11) and(24)0>Ψ1+Ψ2+Ψ3+Ψ4+Ψ5+Ψ6,where {Ψi}i=1,…,6 are defined in (13), (15), (18), (20), (26), and (27), respectively.Proof.
Let us redefine the augmented stateζk as(25)ζkT=xkTxk-d1Txk-d2Txk-d3TΔxkTΦ011xiTΦ121xiTΦ231xiTpkTand establish a block entry matrix e8 such that pk=e8ζk. Then, the term Ψ5 is naturally converted into(26)Ψ5=Hee0TZ0+e4TZ4e4+I-Ae0-Ade2-Ge8,and the uncertainty such that 0≤qkTqk-pkTpk is represented as 0≤ζkTΨ6ζk, where(27)Ψ6=Ee0+Ede2TEe0+Ede2-e8Te8.As a result, ΔVk≤ζkT(Ψ1+⋯+Ψ5+Ψ6)ζk; thus, the robust stability criterion is given by (11) and (24).


## 4. Control Synthesis

Let us consider a linear system of the following form:(28)xk+1=Axk+Buk-dk,where uk-d(k)∈Rnu denotes the delayed control input. Then, under the state-feedback control law uk=Fxk, the closed-loop control system is described as follows:(29)xk+1=Axk+Adxk-dk,where Ad=BF and F denotes the control gain to be designed.Theorem 7.
Letd1=d_, d3=d¯, ϵ be prescribed and define δ=d3-d1. The closed-loop system in (29) is asymptotically stable for any time-varying d(k) satisfying d(k)∈[d_,d¯], if there exist matrices {S¯i}i=1,…,4∈Rnx×nx, W¯∈Rnx×nx, and F¯∈Rnu×nx and symmetric matrices P¯∈R3nx×3nx, {Q¯i}i=1,2∈Rnx×nx, {R¯1i}i=1,2∈Rnx×nx, {R¯2i}i=1,2∈Rnx×nx, and {X¯i}i=0,1,2∈Rnx×nx such that(30)
0
>
Ψ
¯
1
+
Ψ
¯
2
+
Ψ
¯
3
+
Ψ
¯
4
+
Ψ
¯
5
,
(31)
0
≤
R
¯
21
X
¯
1
S
¯
1
S
¯
2
∗
R
¯
22
+
X
¯
1
S
¯
3
S
¯
4
∗
∗
R
¯
21
X
¯
2
∗
∗
∗
R
¯
22
+
X
¯
2
,where {Ψ¯i}i=1,…,5 are defined in (34)–(38), respectively. Moreover, the control gain F can be reconstructed by F=F¯W¯-1.Proof.
First of all, let us consider a nonsingular matrixW¯ of the following form:(32)W¯=e0TW¯e1TW¯e2TW¯e3TW¯e4TW¯e5TW¯e6TW¯e7TW¯,which satisfies that eiW¯=W¯ei for all i. Further, define(33)P¯=W¯000W¯000W¯TPW¯000W¯000W¯=P¯1P¯2P¯3∗P¯4P¯5∗∗P¯6and let Z0=W and Z4=ϵW, where W=W¯-1, and ϵ is a scalar variable. Then, pre- and postmultiplying Ψi by W¯T and W¯ yields Ψ¯i=W¯TΨiW¯:(34)Ψ¯1=e4+e0e0-e1+e5e1-e3+e6+e7TP¯e4+e0e0-e1+e5e1-e3+e6+e7-e0e5e6+e7TP¯e0e5e6+e7=e4TP¯1e4+Hee4TP¯1e0+e4TP¯2e0-e1+e4TP¯2e5+e4TP¯3e1-e3+e4TP¯3e6+e7+Hee0TP¯2e0-e1+e0TP¯3e1-e3+e0-e1TP¯4e0-e1+Hee0-e1TP¯4e5+e0-e1TP¯5e1-e3+e0-e1TP¯5e6+e7+Hee5TP¯5e1-e3+e1-e3TP¯6e1-e3+Hee1-e3TP¯6e6+e7,(35)Ψ¯2=e0TW¯TQ1W¯︸=Q¯1e0-e1TQ¯1e1+e0TW¯TQ2W¯︸=Q¯2e0-e3TQ¯2e3,(36)Ψ¯3=d12e0TW¯TR11W¯︸=R¯11e0+e4TW¯TR12W¯︸=R¯12e4+d1e0TW¯TX0W¯︸=X¯0e0-d1e1TX¯0e1-e5TR¯11e5-Hee5TX¯0e0-e1-e0-e1TR¯12e0-e1-e0-e1TX¯0e0-e1,(37)Ψ¯4=δ2e0TW¯TR21W¯︸=R¯21e0+e4TW¯TR22W¯︸=R¯22e4+δe1TW¯TX1W¯︸=X¯1e1-δe2TX¯1e2+δe2TW¯TX2W¯︸=X¯2e2-δe3TX¯2e3-e6TR¯21e6-Hee6TX¯1e1-e2+e6TW¯TS1W¯︸=S¯1e7-Hee6TW¯TS2W¯︸=S¯2e2-e3-e1-e2TR¯22e1-e2-e1-e2TX¯1e1-e2-Hee1-e2TW¯TS3W¯︸=S¯3e7+e1-e2TW¯TS4W¯︸=S¯4e2-e3-e7TR¯21e7-Hee7TX¯2e2-e3-e2-e3TR¯22e2-e3-e2-e3TX¯2e2-e3,(38)Ψ¯5=HeW¯Te0TZ0+e4TZ4W¯e4+I-AW¯e0-AdW¯e2=Hee0T+ϵe4TW¯e4+W¯-AW¯e0-BFW¯︸=F¯e2.In other words, the stabilization condition is given by(39)0>∑i=15W¯TΨiW¯=∑i=15Ψ¯i,which becomes (30). Here, (30) implies e7(∑i=15‍Ψ¯i)e7T=-R¯21=-W¯TR21W¯<0; that is, W¯ is nonsingular and thus W¯ becomes also nonsingular. Next, pre- and postmultiplying (11) by diag(W¯T,W¯T,W¯T,W¯T) and its transpose yields (31).


## 5. Numerical Examples

Three numerical examples are considered in order to illustrate the effectiveness of the obtained results.Example 1 (stability analysis).
Let us consider a delayed discrete-time system used in [17]:(40)xk+1=0.800.000.050.90xk+-0.10.0-0.2-0.1xk-dk,where d_≤d(k)≤d¯. For (40) with various d_, Table 1 lists the maximum allowable upper bounds (MAUBs) of d(k), obtained by Theorem 2 and different methods. From Table 1, it can be seen that the stability criteria established in [10, 11] and Theorem 2 offer the most improved of the results. In particular, it is noteworthy that Theorem 2 provides the same delay bounds (i.e., stability region) as those of [10, 11] under the requirement of much less computational complexity with respect to the number of scalar variables (NSVs), as mentioned in Remark 3. That is, in contrast with [10, 11, 24], Theorem 2 offers a more efficient approach in terms of both performance and computational complexity.Table 1
Example1: MAUB of d(k) for various d_.
d
_
2
6
10
15
20
NSV
Theorem1 [17]
13
14
17
21
25
21
Theorem1 [18]
14
16
18
21
25
18
Proposition1 [23]
17
18
20
23
27
38
Corollary1 [24]
19
20
21
24
27
82
Corollary3 [11]
22
22
23
25
28
391
Theorem2 [10]
22
22
23
25
28
126
Theorem2
22
22
23
25
28
72Example 2 (robust stability analysis).
Consider the following uncertain discrete-time system with time-varying delay used in [24]:(41)xk+1=0.80.00.00.9xk+-0.10.0-0.1-0.1xk-dk+α0pk,qk=1.00.0xk.For each d_≤d(k)≤d¯, the MAUBs of α such that (23) is robustly asymptotically stable are listed in Table 2, obtained by Corollary 6 and different methods. That is, from Table 2, we can see that the robust stability criterion given in Corollary 6 is much more efficient than the ones in [24] from the viewpoint of both performance and computational complexity.Table 2
Example2: MAUB of α for various d(k)∈[d_,d-].
d
(
k
)
∈
[
d
_
,
d
-
]
[
2,7
]
[
5,10
]
[
10,15
]
[
20,25
]
NSV
Theorem1 [24]
0.2007
0.1554
0.1144
0.0957
82
Corollary6
0.2013
0.1555
0.1155
0.0961
72Example 3 (control synthesis).
Consider the following discrete-time system transformed from the continuous-time model of an inverted pendulum (refer to [7]):(42)xk+1=1.00780.03010.52021.0078xk+-0.0001-0.0053uk-dk.The goal of this example is to design the control uk=Fxk that stabilizes (42) with 1≤d(k)≤d¯, such that the closed-loop system is asymptotically stable. Table 3 shows the MAUBs of d(k) and the corresponding control gains, obtained by Theorem 7 (ϵ=100). From Table 3, we can see that the proposed stabilization condition is significantly valuable in the sense that it requires less computational complexity as well as providing larger MAUB than that of [26]. Meanwhile, based on the obtained control gain, Figure 1 shows the state responses of (42) with xk=1-1T for k∈{-7,…,0} and d(k)=6·⌈sin(πk/2)⌉+1∈[1,7]. Here, it can be found that the state converges to zero as time goes to infinity.Table 3
Example3: MAUB of d(k) and control gains F when d_=1.
Methods
d
-
Control gains
NSV
Theorem3 [26]
4
F
=
110.6827
34.6980
BMI problem
Theorem3 [10]
7
F
=
98.5858
24.0621
131
Theorem7
7
F
=
100.5577
24.4503
70Figure 1
State responsesxk=[x1,kx2,k]T and time-varying delay d(k)=6⋅⌈sin(πk/2)⌉+1.Remark 8.
In Example3, the matrix W is given by(43)W=W¯-1=-24.0367-5.9185-5.7376-1.5868,forϵ>0.Thus, as mentioned in Remark 4, the following relation is satisfied: He[Z4]=He[ϵW]<0.


## 6. Concluding Remarks

In this paper, the problem of deriving an efficient stability criterion is investigated for discrete-time systems with time-varying delay. The main feature herein is that the conservatism of a stability criterion is reduced in spite of the requirement of less computational complexity. In addition, the stabilization problem of systems with time-delayed control input is addressed in the LMI framework.



---

*Source: 813123-2015-10-11.xml*