Delayed Discrete-Time Systems Stability Analysis

Automatica 147 (2023) 110592
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Automatica
journal homepage: www.elsevier.com/locate/automatica
Technical communique
Stability analysis of delayed discrete-time systems based on a
delay-square-dependent Lyapunov functional✩
Xian-Ming Zhang a, Qing-Long Han a,∗, Xiaohua Ge a, Chen Peng b
aSchool of Science, Computing and Engineering Technologies, Swinburne University of Technology, Melbourne VIC 3122, Australia
bSchool of Mechatronic Engineering and Automation, Shanghai University, Shanghai 200444, China
article info
Article history:
Received 31 July 2021
Received in revised form 17 May 2022
Accepted 15 July 2022
Available online 10 September 2022
Keywords:
Discrete-time systems
Time-varying delay
Stability
Lyapunov functional
Quadratic polynomial matrix
abstract
Recent research has shown that one can obtain a less conservative stability criterion for a continuous-
time linear system with a time-varying delay by introducing a Lyapunov-Krasovskii functional with a
polynomial matrix on the time-varying delay. This paper aims at analysing the stability of discrete-
time linear systems with time-varying delays by introducing a delay-square-dependent Lyapunov
functional. A novel convex method is presented to formulate a less conservative stability criterion,
which is demonstrated through numerical simulation. Moreover, it is also shown that, if the polynomial
inequality method is employed, the resultant stability criterion is inapplicable due to its extremely high
numerical complexity.
©2022 Elsevier Ltd. All rights reserved.
1. Introduction
Discrete-time systems have a strong application background
in modern engineering since a real system is usually imple-
mented in the framework of digital-signal models. Networked
control systems have been a typical example of the applications of
discrete-time systems (Zhang et al.,2022). It has been shown that
time-delays are unavoidable during the digital implementation
of an engineering system. Thus, extensive attention has been
paid to discrete-time systems with time-varying delay in the past
decades, see, e.g. Gu et al. (2003), Kao (2012) and Xu et al. (2005).
As one of fundamental issues of delayed discrete-time sys-
tems, stability has been studied for a long time with a number of
notable methods proposed. To mention a few, in the early 2000s,
a free-weighting matrix approach and a Jensen-like finite-sum
inequality approach were proposed to obtain some nice stability
criteria, by which one can check how big the time-delay is al-
lowed to retain the stability of a delayed discrete-time system. In
2013, a Wirtinger-based integral inequality (WII) was introduced
in Seuret and Gouaisbaut (2013). Inspired by the idea of WII,
several versions of the Jensen-like finite-sum inequality were in-
troduced, such as a summation inequality (Seuret et al.,2015), an
✩The material in this paper was not presented at any conference. This paper
was recommended for publication in revised form by Associate Editor Rifat
Sipahi under the direction of Editor André Tits.
∗Corresponding author.
E-mail addresses: [email protected] (X.-M. Zhang),
[email protected] (Q.-L. Han), [email protected] (X. Ge), [email protected]
(C. Peng).
auxiliary-function-based discrete inequalities (Nam et al.,2015),
and so on Zhang and Han (2015). These finite-sum inequalities
contribute to less conservative stability conditions for discrete-
time systems with time-varying delay.
Recently, a polynomial inequality approach is developed for
stability analysis of continuous-time systems with time-varying
delay (Zhang et al.,2021). Recalling Proposition 1 in Zhang et al.
(2021), this approach suggests a Lyapunov–Krasovskii functional
with a quadratic polynomial matrix. Then the time-derivative of
the functional along with the trajectory of the system is estimated
as a cubic polynomial on the time-varying delay. As a result, a less
conservative stability criterion is obtained using a sufficient con-
dition on cubic polynomial inequalities. Clearly, the polynomial
inequality approach opens a brand new door to analyse stability
of time-delay systems. A natural question arises: Is the polynomial
inequality approach applicable to discrete-time systems with time-
varying delay? To answer this question is of significance in theory
and in practice, which is the motivation of this technical note.
It should be mentioned that it is not a trivial extension of
the polynomial inequality approach in Zhang et al. (2021) from
continuous-time systems to discrete-time systems. In fact, fol-
lowing the key idea in Zhang et al. (2021), a delay-dependent
matrix, P(t)≜ℓ2(t)P2+ℓ(t)P1+P0is introduced in the Lyapunov
functional, where Pi(i=0,1,2) are constant real symmetric
matrices and ℓ(t) is the time-varying delay. Clearly, ˙ P(t) is linear
on the delay variation rate ˙ ℓ(t). However, if the counterpart
P(k) of P(t) is introduced into a discrete-time system, ∆P(k)
is nonlinear on the delay variation rate ∆ℓ(k). In this paper, we
provide an insightful analysis to the question above. First, a novel
https://doi.org/10.1016/j.automatica.2022.110592
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X.-M. Zhang, Q.-L. Han, X. Ge et al. Automatica 147 (2023) 110592
Lyapunov functional that includes a delay-square-dependent Lya-
punov matrix P(k) is constructed. Second, two methods, namely
a convex method and a polynomial inequality method, are em-
ployed to obtain two stability criteria for discrete-time linear
systems with time-varying delay. A comparison between their
numerical complexities shows that the stability criterion ob-
tained by the polynomial inequality method is unworkable due
to its extremely high numerical complexity. However, the stabil-
ity criterion derived from the convex method can achieve less
conservative results than some existing ones, which is verified
through a numerical example.
Notations: He{M} = M+MT; col{· · · } means a column (or block-
column) vector; and col{[A⋆],[B C]} stands for col{[A BT],[B C]}.
Sn(Sn
+) means the set of symmetric (positive definite) matrices
of Rn×n.
2. Problem formulation and lemmas
Consider the following discrete-time system
x(k+1) =Ax(k)+Bx(k−ℓ(k))
x(θ)=φ(θ), θ = −ℓ2,−ℓ2+1,...,0.(1)
where x∈Rnis the state vector, and Aand Bare n×n
known real matrices, φ(θ) is the initial condition. Suppose that
the time-varying delay function τ(k) satisfies
1≤ℓ1≤ℓ(k)≤ℓ2,d1≤∆ℓ(k)≤d2,(2)
for k=1,2, . . ., where ∆ℓ(k)=ℓ(k+1) −ℓ(k), and ℓ1, ℓ2,d1and
d2are known integers.
Stability of the time-delay system (1) has been studied for
a long time using the Lyapunov functional method. By con-
structing a proper Lyapunov functional positive-definite, a delay-
dependent stability criterion can be obtained from its forward
difference negative-definite. Recalling some existing results on
this issue, there are a number of Lyapunov functionals con-
structed. In order to seek less conservative stability criteria, this
paper introduces a novel Lyapunov functional, which reads as
V(k)=V1(k)+V2(k)+V3(k)+V4(k) (3)
where
V1(k)=χT
1(k)P(k)χ1(k),
V2(k)=k−1
j=k−ℓ1χT
2(k,j)Q1χ2(k,j)
+k−ℓ1−1
j=k−ℓ2χT
3(k,j)Q2χ3(k,j),
V3(k)=ℓ1−1
j=−ℓ1k−1 i=k+jδT(i)R1δ(i)
+(ℓ2−ℓ1)−ℓ1−1
j=−ℓ2k−1
i=k+jδT(i)R2δ(i),
V4(k)=(ℓ2−ℓ1)−ℓ1−1
j=−ℓ2k−1
i=k+jβT(i)R3β(i),
with
P(k)=ℓ2(k)P2+ℓ(k)P1+P0,(4)
δ(k)=x(k+1) −x(k), β(k)=col{x(k), δ(k)},(5)
χ1(k)=colx(k),k−1
j=k−ℓ1x(j),k−ℓ1−1
j=k−ℓ2x(j),
χ2(k,j)=colx(k),x(j),k−1
i=jx(i),j
i=k−ℓ1x(i),
χ3(k,j)=colx(k),x(j),k−ℓ1−1
i=jx(i),j i=k−ℓ2x(i).
The novelty of the Lyapunov functional V(k) lies in that a quadratic
polynomial matrix P(k) on ℓ(k) is introduced rather than a con-
stant matrix P. However, the introduction of P(k) makes the
stability analysis more complicated. The problem to be addressed
in this paper is then stated as
OP1: How to formulate a stability criterion based on the Lyapunov
functional V (k)in (3)?
In the next section, we provide solutions to the problem OP1.
For this goal, we need the following lemmas.
Lemma 1 (Nam et al.,2015).For a given n ×n real matrix R >0,
integers r1and r2with r2>r1, and an n-dimension real vec-
tor sequence {x(r2),...,x(r1)}, the following finite-sum inequalities
hold
¯
hr2−1
j=r1δT(j)Rδ(j)≥ϖT
0Rϖ0+3ϖT 1Rϖ1+5ϖT 2Rϖ2,
¯
hk−r1−1
j=k−r2xT(j)Rx(j)≥ ¯ϖT
0R¯ϖ0+3¯ϖT 1R¯ϖ1,
where δ(k)is given in (5),¯ h:= r2−r1, and
ϖ0:= x(r2)−x(r1),
ϖ1:= x(r1)+x(r2)−2
¯
h+1r2
j=r1x(j),
ϖ2:= ϖ0+6
(¯ h+1)(¯ h+2) r2
j=r1(r2+r1−2j)x(j),
¯ϖ0:= k−r1
j=k−r2x(j)−x(k−r1),
¯ϖ1:= x(k−r1)−1
¯
h+1k−r1
j=k−r2x(j)
+k−r1
j=k−r2
2k−r1−r2−2j
¯
h+1x(j).
Lemma 2 (Zhang et al.,2021).The polynomial matrix f3(s)=s3Φ3+
s2Φ2+sΦ1+Φ0<0,∀s∈ [h1,h2], if there exists a constant
compatible real matrix M such that
Ω(h1)+He{MB(h1)}<0,
Ω(h2)+He{MB(h2)}<0,
where
Ω(s)=sΦ1+Φ0s
2Φ2
s
2Φ2sΦ3,B(s)= [sI −I].
3. Main results
In this section, we study the stability of the system (1) by
employing two methods: a convex method and a polynomial in-
equality method (Zhang et al.,2021). To begin with, we denote
θ1k:= ℓ(k)−ℓ1+1, θ2k:= ℓ2−ℓ(k)+1,
θ01 =ℓ1+1, θ02 =ℓ2−ℓ1,g(a):= 1
2a(a+1),
℘1k:= 1
θ01 k
i=k−ℓ1x(i), ℘2k:= 1
θ1kk−ℓ1
i=k−ℓ(k)x(i),
℘3k:= 1
θ2kk−ℓ(k)
i=k−ℓ2x(i), ℘4k:= k
i=k−ℓ1
2k−ℓ1−2i
2g(θ01)x(i),
℘5k:= k−ℓ1
i=k−ℓ(k)
2k−ℓ1−ℓ(k)−2i
2g(θ1k)x(i),
℘6k:= k−ℓ(k)
i=k−ℓ2
2k−ℓ(k)−ℓ2−2i
2g(θ2k)x(i).
3.1. Stability criterion based on a convex method
First we introduce a vector as
ξ(k):= col x(k+1),x(k),x(k−ℓ(k)),x(k−ℓ1),
x(k−ℓ2), ℘1k, ℘2k, ℘3k, ℘4k, ℘5k, ℘6k, θ1k℘2k,
θ2k℘3k,(θ1k+1)℘5k,(θ2k+1)℘6k},(6)
and ej(j=1,2,...,15) are row-block vectors such that x(k+
1) =e1ξ(k),...,(θ2k+1)℘6k=e15ξ(k).
Proposition 3. For given scalars ℓ1, ℓ2,d1and d2, the system (1)
is asymptotically stable if there exist real matrices Pj∈S3n(j=
0,1,2),Qm∈S4n
+, Rm∈Sn +, R3∈S2n +, Nm∈R3n×3n, Tm∈Sn,
2

X.-M. Zhang, Q.-L. Han, X. Ge et al. Automatica 147 (2023) 110592
X1∈R15n×6n,X2∈R6n×6n,X3∈R15n×n,X4∈R15n×4nand
Ym∈R7n×7n(m=1,2) such that for d ∈ {d1,d2}and ℓ∈ {ℓ1, ℓ2}
ℓP1+P0+He{ℓN1}⋆
ℓ(P2/2+NT
2)−N1−He{N2}>0,(7)
Ti=R3+Ti>0,Ti≜0Ti
TiTi,i=1,2,(8)
Ξ(ℓ2,d)⋆ ⋆
−XT
1+X2ℵ13 Ξ12(ℓ2,d)⋆
YT
1ℵ43 0−Y1
<0,(9)
Ξ(ℓ1,d)⋆ ⋆
−XT
1+X2ℵ13 Ξ12(ℓ1,d)⋆
Y2ℵ44 0−Y2
<0,(10)
where
Yi=diag{ Ti,3 Ti,R2,3R2,5R2},i=1,2,(11)
Ξ(ℓ(k),∆ℓ(k)) =Ξ11 +Ξ2+Ξ3,(12)
Ξ11 = ℵT
12[(ℓ(k)+∆ℓ(k))P1+P0]ℵ12 +He{X1ℵ13}
− ℵT
11[ℓ(k)P1+P0]ℵ11,(13)
Ξ12 =diag{P2,−P2−He{X2},(14)
Ξ2= ℵT
23Q1ℵ23 − ℵT 24Q1ℵ24 + ℵT 26Q2ℵ26 − ℵT 27Q2ℵ27
+ℓ1ℵT
21Q1ℵ21 +θ02ℵT 22Q2ℵ22
+He{ℵT
21Q1ℵ25 + ℵT 22Q2ℵ28},(15)
Ξ3=(e1−e2)T(ℓ2
1R1+θ2 02R2)(e1−e2)+θ2 02ℵT 40R3ℵ40
− ℵT
31R1ℵ31 −(2 −α)ℵT 43Y2ℵ43 −(1 +α)ℵT 44Y1ℵ44
+HeX3ℵ0+X4ℵ51 − ℵT
43[αY1+(1 −α)Y2]ℵ44
+θ02eT
4T1e4+eT 3(T2−T1)e3−eT 5T2e5,(16)
with ℵ0=e1−Ae2−Be3, and
ℵ11 =col{e2, θ01e6−e2,ℑ2−e4},ℑ2=e12 +e13 −e3,
ℵ12 =col{e1, θ01e6−e4,ℑ2−e5},ℵ40 =col{e2,e1−e2},
ℵ13 =col{(ℓ(k)+∆ℓ(k)))ℵ12, ℓ(k)ℵ11},
ℵ21 =col{e1−e2,0,e2,−e4},R1=diag{R1,3R1,5R1},
ℵ22 =col{e1−e2,0,e4,−e5},ℵ23 =col{e2,e2,0, θ01e6},
ℵ24 =col{e2,e4, θ01e6−e2,e4},ℵ26 =col{e2,e4,0,ℑ2},
ℵ25 =col{ℓ1e2, θ01e6−e4,ℑ1,g(θ01)(e6+e9)−e4},
ℵ27 =col{e2,e5,ℑ2−e4,e5},ℵ28 =col{θ02e2,ℑ2−e5,
ℓ2ℑ2− ℑ3−θ02e4,ℑ3−(ℓ1−1)ℑ2−e5},
ℵ31 =col{e2−e4,e2+e4−2e6,e2−e4+6e9},
ℵ32 =col{e3−e5,e3+e5−2e8,e3−e5+6e11},
ℵ33 =col{e4−e3,e4+e3−2e7,e4−e3+6e10},
ℵ41 =col{e13 −e3,e3−e5,e3−e8+e15,−e3−e5+2e8},
ℵ42 =col{e12 −e4,e4−e3,e4−e7+e14,−e3−e4+2e7},
ℵ51 =col{e12 −θ1ke7,e13 −θ2ke8,e14 −(θ1k+1)e10,
e15 −(θ2k+1)e11},ℵ43 =col{ℵ41,ℵ32},
ℑ1=g(θ01)(e6−e9)−θ01e6−ℓ1e2,ℵ44 =col{ℵ42,ℵ33},
ℑ3=0.5[θ1ke14 +θ2ke15 +ℓ(k)(ℑ2−e3)+ℓ1e12 +ℓ2e13].
Proof. In order to streamline the proof, some detailed calcula-
tions are omitted due to page limitation. We complete the proof
by three steps.
Step 1:V(k) in (3) is positive-definite. In fact, the conditions in (7)
ensure P(k)>0 for ℓ1≤ℓ(k)≤ℓ2by Lemma 2. Thus, it is easy
to verify that the Lyapunov functional V(k) is positive-definite.
Step 2: Calculate the forward difference ∆V(k)=V(k+1)−V(k)=
4
j=1∆Vj(k).
For ∆V1(k), we introduce two vectors as
ζ1(k)=ℓ(k+1)χ1(k+1), ζ2(k)=ℓ(k)χ1(k).
Then we have
χT
1(k+1)P(k+1)χ1(k+1) =ζT 1(k)P2ζ1(k)
+χT
1(k+1)[ℓ(k+1)P1+P0]χ1(k+1),
χT
1(k)P(k)χ1(k)=ζT 2(k)P2ζ2(k)
+χT
1(k)[ℓ(k)P1+P0]χ1(k).
It is not difficult to verify that
χ1(k)= ℵ11ξ(k), χ1(k+1) = ℵ12ξ(k).
Thus, ∆V1(k) reads as
∆V1(k)=ξT(k)ℵT
12[ℓ(k+1)P1+P0]ℵ12ξ(k)
−ξT(k)ℵT
11[ℓ(k)P1+P0]ℵ11ξ(k)
+ζT
1(k)P2ζ1(k)−ζT 2(k)P2ζ2(k).
Note that ζ1(k)=ℓ(k+1)ℵ12ξ(k), ζ2(k)=ℓ(k)ℵ11ξ(k),which
lead to
0=ℓ(k+1)ℵ12
ℓ(k)ℵ11 ξ(k)−ζ1(k) ζ2(k)= ℵ13ξ(k)−ζ1(k) ζ2(k),
where ℵ13 =col{(ℓ(k)+∆ℓ(k))ℵ12, ℓ(k)ℵ11}. Let ζ(k)=col{ζ1(k),
ζ2(k)}. Then for real matrices X1and X2with appropriate dimen-
sions, the following holds
0=2[ξT(k)X1+ζT(k)X2][ℵ13ξ(k)−ζ(k)],
which follows that
∆V1(k)=ξ(k)
ζ(k)TΞ11 ⋆
X2ℵ13 −XT
1Ξ12ξ(k)
ζ(k),(17)
where Ξ11 and Ξ12 are defined in (13) and (14), respectively. To
calculate ∆V2(k), we first note that
χ2(k+1,j)=χ2(k,j)+ ℵ21ξ(k),
χ3(k+1,j)=χ3(k,j)+ ℵ22ξ(k),
and
χ2(k,k)= ℵ23ξ(k), χ2(k,k−ℓ1)= ℵ24ξ(k),
k
j=k+1−ℓ1χ2(k,j)= ℵ25ξ(k), χ3(k,k−ℓ1)= ℵ26ξ(k),
χ3(k,k−ℓ2)= ℵ27ξ(k),k−ℓ1
j=k+1−ℓ2χ3(k,j)) = ℵ28ξ(k).
Then some algebraic manipulations give
∆V2(k)=ξT(k)Ξ2ξ(k) (18)
where Ξ2is given in (15).
For ∆V3(k) and ∆V4(k), noting that from Kwon et al. (2012),
one has
0=θ02ξT(k)[eT
4T1e4+eT 3(T2−T1)e3−eT 5T2e5]ξ(k)
−θ02k−ℓ(k)−1
j=k−ℓ2βT(j)T2β(j)
−θ02k−ℓ1−1
j=k−ℓ(k)βT(j)T1β(j),
3

X.-M. Zhang, Q.-L. Han, X. Ge et al. Automatica 147 (2023) 110592
where Tiis defined in (8). Then by Lemma 1, one gets
∆V3(k)+∆V4(k)≤ξT(k)(Ξ31 +Ξ32 +θ02ℵT
40R3ℵ40)ξ(k)
−1
αξT(k)ℵT
41diag{
T2,3 T2}ℵ41 + ℵT
32R2ℵ32ξ(k)
−1
1−αξT(k)ℵT
42diag{
T1,3 T1}ℵ42 + ℵT
33R2ℵ33ξ(k),(19)
where α=(ℓ2−ℓ(k))/(ℓ2−ℓ1), and
Ξ31 =(e1−e2)T(ℓ2
1R1+θ2 02R2)(e1−e2)− ℵT 31R1ℵ31
Ξ32 =θ02eT
4T1e4+eT 3(T2−T1)e3−eT 5T2e5,
R2=diag{R2,3R2,5R2}, Ti=R3+Ti(i=1,2).
For the last two terms in (19), by employing the improved
reciprocally convex inequality, we obtain
−1
αℵT
41diag{
T2,3 T2}ℵ41 + ℵT
32R2ℵ32
−1
1−αℵT
42diag{
T1,3 T1}ℵ42 + ℵT
33R2ℵ33
≤ −ℵT
43[Y2+(1 −α)(Y2−Y1Y−1
1YT
1)]ℵ43
− ℵT
44[Y1+α(Y1−YT 2Y−1
2Y2)]ℵ44
−HeℵT
43[αY1+(1 −α)Y2]ℵ44,(20)
where Y1and Y2are given in (11). Substituting (20) to (19) gives
∆V3(k)+∆V4(k)≤ξT(k)Ξ3−He{X3ℵ0+X4ℵ51}
+(1 −α)ℵT
43Y1Y−1
1YT
1ℵ43
+αℵT
44YT 2Y−1
2Y2ℵ44ξ(k),(21)
where Ξ3is given in (16). On the other hand, from the system
(1), it is clear that
0=x(k+1) −Ax(k)−Bx(k−ℓ(k)) = ℵ0ξ(k).
And the last four entries in ξ(k)(12) satisfy
θ1k℘2k=e12ξ(k)=θ1ke7ξ(k),
θ2k℘3k=e13ξ(k)=θ2ke8ξ(k),
(θ1k+1)℘5k=e14ξ(k)=(θ1k+1)e10ξ(k),
(θ2k+1)℘6k=e15ξ(k)=(θ2k+1)e11ξ(k),
which lead to
0= ℵ51ξ(k).
Employing the free-weighting matrix approach, for X3∈R15n×n
and X4∈R15n×4n, one has
0=2ξT(k)X3ℵ0ξ(k),0=2ξT(k)X4ℵ51ξ(k).(22)
Based on (17),(18),(21) and (22), we have
∆V(k)≤ ξT(k)Υ(ℓ(k),∆ℓ(k)) ξ(k),(23)
where  ξ(k)=col{ξ(k), ζ (k)}and
Υ(ℓ(k),∆ℓ(k)) =3
i=1Ξi+(1 −α)Λ1+αΛ2⋆
−XT
1+X2ℵ13 Ξ12,(24)
Λ1= ℵT
43Y1Y−1
1YT
1ℵ43,Λ2= ℵT 44YT 2Y−1
2Y2ℵ44.
Step 3: Prove the conclusion. In fact, if the matrix inequalities in
(9) and (10) are satisfied, then using the Schur complement gives
Υ(ℓ(k),∆ℓ(k)) <0 for ℓ1≤ℓ(k)≤ℓ2and d1≤∆ℓ(k)≤d2.
Therefore, there exists a sufficiently small scalar ε > 0 such that
∆V(k)≤ −εξT(k)ξ(k)≤ −εxT(k)x(k), which concludes that the
system (1) is asymptotically stable. □
Remark 4. From the proof of Proposition 3, the forward differ-
ence ∆V(k) is estimated in (23), where the matrix Υ(ℓ(k),∆ℓ(k))
is linear on both ℓ(k) and ∆ℓ(k). This contributes to the treat-
ment of ∆V1(k)in(17) and the introduction of the vectors
θ1k℘2k, θ2k℘3k,(θ1k+1)℘5k,(θ2k+1)℘6kinto ξ(k) in (6). Thus, the
convex method can be employed to formulate a stability criterion.
3.2. Stability criterion based on the polynomial inequality method
In this subsection, following the idea in Zhang et al. (2021), we
present a stability criterion for the system (1).
For the forward difference of V1(k), it is clear that
∆V1(k)=χT
1(k+1)P(k+1)χ1(k+1)
−χT
1(k)P(k)χ1(k).(25)
Let ϖ(k)=col{x(k+1),x(k),x(k−ℓ(k)),x(k−ℓ1),x(k−
ℓ2), ℘1k, ℘2k, ℘3k, ℘4k, ℘5k, ℘6k}, and denote
ג1(ℓ(k)) =0I0 0 0 0 0 0 0 0 0
0−I0 0 0 θ01I0 0 0 0 0
0 0 −I−I0 0 θ1kIθ2kI000
ג2=I−I0 0 0 000000
0I0−I0 000000
0 0 0 I−I000000
ℸ(k)= [∆ℓ(k)]2P2+∆ℓ(k)[P1+2ℓ(k)P2].
Then P(k+1) =P(k)+ℸ(k), χ1(k)=ג1(ℓ(k))ϖ(k) and
χ1(k+1) =χ1(k)+ג2ϖ(k)= [ג1(ℓ(k)) +ג2]ϖ(k)
which follow that
∆V1(k)=ϖT(k)ℶ(ℓ(k),∆ℓ(k))ϖ(k) (26)
where
ℶ(ℓ(k),∆ℓ(k)) = [ג1(ℓ(k)) +ג2]Tℸ(k)[ג1(ℓ(k)) +ג2]
+גT
2P(k)ג2+He{גT 2P(k)ג1(ℓ(k))}.(27)
It should be mentioned that ℶ(ℓ(k),∆ℓ(k)) is a matrix-valued
binary polynomial on (ℓ(k),∆ℓ(k)), which is cubic on ℓ(k) and
quadratic on ∆ℓ(k).
The treatments of ∆V2(k),∆V3(k) and ∆V4(k) are similar to
that in the proof of Proposition 3. As such, the forward difference
of V(k) can be estimated as
∆V(k)≤ϖT(k)F(ℓ(k),∆ℓ(k))ϖ(k),(28)
where
F(ℓ(k),∆ℓ(k)) =
3
j=0
ℓj(k)
2
i=0
(∆ℓ(k))i¯ Φji (29)
with ¯ Φji(i=0,1,2,j=0,1,2,3) being real symmetric ma-
trices irrespective of ℓ(k) and ∆ℓ(k). Due to page limitation,
the matrices ¯ Φji(i=0,1,2,j=0,1,2,3) are not given here.
Since F(ℓ(k),∆ℓ(k)) is a matrix-valued binary polynomial on
(ℓ(k),∆ℓ(k)), applying the polynomial inequality (i.e. Lemma 2)
twice, one can readily establish the following result.
Proposition 5. For given scalars ℓ1, ℓ2,d1and d2, the system (1)
is asymptotically stable if there exist real matrices Pj∈S3n(j=
0,1,2),Qm∈S4n
+, Rm∈Sn +, R3∈S2n +, Nm∈R3n×3n, Tm∈Sn, X3∈
R11n×nand M1∈R22n×11n,M2∈R44n×22n,Ym∈R7n×7n(m=1,2)
such that (7),(8) and Γ(ℓ, d)<0for ℓ∈ {ℓ1, ℓ2}and d ∈ {d1,d2},
where
Γ(ℓ, d)=Ω0(ℓ)1
2Ω1(ℓ)
1
2Ω1(ℓ)Ω2(ℓ)+He{M2[dI −I]},(30)
4

X.-M. Zhang, Q.-L. Han, X. Ge et al. Automatica 147 (2023) 110592
Table 1
The admissible upper bound ℓ2for different values of ℓ1.
ℓ15 7 11 13 NoDVs
Nam et al. (2015) 20 22 26 27 29.5n2+12.5n
Chen et al. (2016) 21 22 26 27 78.5n2+12.5n
Zhang et al. (2016) 21 22 26 27 10.5n2+3.5n
Chen et al. (2020) 21 23 27 29 39.5n2+6.5n
Proposition 3 (d=5) 27 29 33 34 341.5n2+8.5n
Proposition 3 (d=2) 28 30 33 35 341.5n2+8.5n
Ωi(ℓ)=ℓ¯
Φ1i+¯ Φ0i1
2ℓ¯
Φ2i
1
2ℓ¯
Φ2iℓ¯ Φ3i,i=1,2,
Ω0(ℓ)=ℓ¯
Φ10 +¯ Φ00 1
2ℓ¯
Φ20
1
2ℓ¯
Φ20 ℓ¯ Φ30 +He{M1[ℓI−I]}.
3.3. Comparison on numerical complexities
From the analysis above, two stability criteria, Propositions 3
and 5, are obtained using a convex method and a polynomial
inequality method, respectively. In the following, we make a
comparison between their numerical complexities.
By Han (2009), the numerical complexity of Proposition 3 or
5is proportional to LM3, where Lis the total row size of the
related matrix inequalities, and Mmeans the total number of
scalar decision variables. We now calculate the pairs (L1,M1)
and (L2,M2) of the total row size Ljand the total number
Mj(j=1,2) of scalar decision variables for Propositions 3 and
5, respectively.
•For Proposition 3:L1=100n,M1=341.5n2+8.5n;
•For Proposition 5:L2=192n,M2=1372.5n2+8.5n.
Thus, the numerical complexity of Proposition 5 is propor-
tional to L2M3
2=192n×(1372.5n2+8.5n)3, which is much
bigger than L1M3
1=100n×(341.5n2+8.5n)3.
For a small number of n=2, the numerical complexity of
Proposition 5 is proportional to L2M3
2=384 ×55073>1.5×
212 ×1010, which is extremely high. Therefore, Proposition 5 is
unworkable to check if the system (1) is asymptotically stable
while Proposition 3 is applicable. In Section 4, through simulation,
one can see that Proposition 3 can deliver larger delay upper
bounds than some existing ones. However, how to reduce the
numerical complexity of Proposition 3 is still challenging.
Remark 6. In this paper, we develop a convex method to study
the stability of delayed discrete-time systems by introducing a
novel Lyapunov functional (3) with a matrix P(k)=ℓ2(k)P2+
ℓ(k)P1+P0. The matrix P(k) can be extended to a general one
like P(k)=s
i=0ℓi(k)Pi, where sis a positive integer. Similar to
the proof of Proposition 3, one can get a stability criterion based
on the Lyapunov functional with P(k). Nevertheless, the resultant
stability criterion should be of higher numerical complexity for a
larger s.
4. Numerical examples
Consider the system (1), where
A=0.648 0.04
0.12 0.654,B=−0.1512 −0.0518 0.0259 −0.1091(31)
This example is widely studied in the literature. For different
values of ℓ1, the admissible upper bounds of ℓ2are listed in
Table 1 by applying some existing stability criteria, e.g. Chen et al.
(2016, Theorem 1), Chen et al. (2020, Theorem 1), Nam et al.
(2015, Theorem1) and Zhang et al. (2016, Theorem1). It should
be pointed out that, those stability criteria are independent of
the delay-variation rate. Thus, they may be conservative when
information on the delay-variation rate is available, that is, when
the bounds d1and d2are known. In fact, if d=d2= −d1∈
{2,5}for this example, Proposition 3 can deliver larger upper
bound ℓ2for the same values of ℓ1, which can be seen in Table 1.
However, the required number of decision variables (NoDVs) in
Proposition 3 is much bigger than those in Chen et al. (2016,
2020), Nam et al. (2015) and Zhang et al. (2016).
On the other hand, the matrix Bin this example is nonsin-
gular. If Bis singular, Proposition 3 is still available to check
the stability of the system. In fact, if we change the matrix B
in (31) to [−0.1512 −0.0518
0 0 ], then for d=2, the admissible upper
bounds of ℓ2are computed as 27,29,32,34 for ℓ1=5,7,11,13,
respectively.
5. Conclusion
Stability of discrete-time systems with time-varying delay has
been investigated in this paper by introducing a novel Lyapunov
functional V(k) with a quadratic polynomial matrix. An effective
method to deal with the forward difference of V(k) has been
presented, leading to a less conservative stability criterion, which
has been illustrated through a numerical example.
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