654 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 64, NO. 2, FEBRUARY 2019
Vector-Lyapunov-Function-Based Input-to-StateStability of Stochastic Impulsive Switched
Time-Delay SystemsWei Ren and Junlin Xiong , Member, IEEE
Abstract—In this paper, the input-to-state stability is stud-ied for stochastic impulsive switched time-delay systems.Using the vector Lyapunov function, average dwell time, andthe properties of M -matrix, different types of sufficient con-ditions are established. Both the case that the continuousdynamics is stable and the case that the discrete dynamicsis stable are addressed, and the stability conditions are ob-tained. In the obtained stability conditions, different compo-nents of the vector Lyapunov function are allowed to be cou-pled; the information in consecutive impulsive switchingintervals is also allowed to be coupled. Therefore, the mag-nification on the corresponding coupling items is avoidedand the obtained results are more general and less conser-vative than the existing results. Furthermore, we investigatethe relationships among the vector Lyapunov function ap-proach, the approach based on the comparison principleand the scalar Lyapunov function approach. According tothe vector Lyapunov function, the comparison system isconstructed and the scalar-Lyapunov-function-based sta-bility conditions are established. Finally, the applicability ofour results is illustrated through two examples from neuralsystems and the synchronization problem of chaos-basedsecure communication systems.
HYBRID systems are dynamic systems that exhibit bothcontinuous and discrete dynamic behaviors [1]. As two
special classes of hybrid systems, impulsive systems andswitched systems, have attracted considerable attention in re-cent years due to their numerous interdisciplinary applicationsin different fields of science and engineering. See [1]–[3] andreferences therein for general introduction and practical applica-tions of impulsive systems and switched systems. Impulsive sys-tems model real-world processes that undergo abrupt changes
Manuscript received October 17, 2017; revised January 6, 2018; ac-cepted April 6, 2018. Date of publication May 14, 2018; date of currentversion January 28, 2019. This work was supported by the NationalNatural Science Foundation of China, under Grant 61374026 and Grant61773357. Recommended by Associate Editor C. Seatzu. (Correspond-ing author: Junlin Xiong.)
The authors are with the Department of Automation, University ofScience and Technology of China, Hefei 230026, China (e-mail: [email protected]; [email protected]).
Color versions of one or more of the figures in this paper are availableonline at http://ieeexplore.ieee.org.
Digital Object Identifier 10.1109/TAC.2018.2836191
(impulses) in the state at discrete times [2], [4]–[7]; switchedsystems model practical systems whose dynamics are chosenfrom a family of possible subsystems based on a switching law[3], [8], [9]. Especially, in some practical systems like Chua’scircuits in [3, Example 1.2.2], both impulsive effects and switch-ing law appear simultaneously in control systems. Such dynamicsystems with both impulsive effects and switching law are calledimpulsive switched systems. Significant contributions on impul-sive switched systems could be found in [3] and [10]–[12] andreferences therein.
In physical systems, external disturbances are inevitable. Boththe continuous dynamics and discrete dynamics may be affected;see survey paper [1]. If the control systems are subjected torandom noises, then stochastic system modeling is required [5],[13], [14]. On the other hand, time delays are frequently encoun-tered in many engineering systems, and may induce oscillation,instability and poor performances [5], [15], [16]. Therefore, amore general and comprehensive system model is proposed, thatis, stochastic impulsive switched time-delay systems. Note thatthere are few works [10], [12] on stochastic impulsive switchedtime-delay systems.
As a fundamental problem of dynamic systems, stabilityproperties have been considered extensively over the past fewdecades. The notation of the input-to-state stability (ISS), orig-inally proposed in [17], has been proven to be useful in charac-terizing the effects of external inputs on a control system. TheISS notation has gradually been extended to different types ofcontrol systems, such as discrete-time systems [18], time-delaysystems [19]–[21], stochastic systems [13], [22], impulsive sys-tems [2], [4]–[6], [15], [23], switched systems [8], [9], [14],[24], [25], and hybrid systems [12], [26], [27]. To study theISS of dynamic systems, a Lyapunov-based approach is com-monly used and various extensions could be found in the lit-erature. For instance, multiple Lyapunov functions were usedin [8], [14], and [24] to study ISS of switched systems. UsingLyapunov–Razumikhin functions and average dwell time, theISS was addressed in [15] for nonlinear impulsive delayed sys-tems. A Lyapunov–Krasovskii functional was applied in [12] tostudy the ISS of impulsive switched delayed systems. GeneralLyapunov functions were used in [6]. However, stability analy-ses in all these works are based on scalar Lyapunov functions.
In the recent years, vector Lyapunov functions, as a viablealternative to scalar Lyapunov functions, have gained increas-ing attention. Vector Lyapunov functions are first introducedin [28], and have been applied to many fields like economics,aerospace engineering, and neural network; see [29, Sec. 4],[30], and [31]. Comparing with scalar Lyapunov functions,
REN AND XIONG: VECTOR-LYAPUNOV-FUNCTION-BASED ISS OF STOCHASTIC IMPULSIVE SWITCHED TIME-DELAY SYSTEMS 655
vector Lyapunov functions have many advantages. In terms ofconstruction of Lyapunov functions, the vector Lyapunov func-tion theory offers more flexible strategies to deal with the com-plexity of dynamic systems; see [29] and [32]–[36]. The reasonlies in that less rigid requirements are imposed on the systemcomponents via the vector Lyapunov function theory; see [29]and [36] for more details. In particular, each component of thevector Lyapunov function only needs to satisfy certain dissipa-tion conditions [32], [33]. This facilitates the choice of vectorLyapunov functions. In terms of stability analysis, the vectorLyapunov function theory allows stability of nonlinear dynamicsystems (especially large-scale systems) to be analyzed via lin-ear system tools; see [37, Sec. 2.2]. In addition, the reduced-dimensional comparison systems can be constructed via vectorLyapunov functions. According to the comparison principle,the stability properties of the comparison systems imply thecorresponding stability properties of the original systems; see[34], [35], [38], and references therein. Hence, the hypothesesfor stability analysis in the scalar Lyapunov function theory areweakened or circumvented by using vector Lyapunov functions,thereby expanding the class of Lyapunov functions that can beused to study the stability of (large-scale) dynamic systems. Inthe literature, there are some works exploring the effectivenessof the vector Lyapunov functions; see [20], [21], [33]–[35], and[38]. For instance, using the vector Lyapunov function approachand comparison principle, feedback stabilization of nonlinearsystems was studied in [32] and[33], ; stability properties of im-pulsive systems were considered in [34], [38], and [39]; stabilityproperties of delayed neural networks were addressed in [20]and [21]. However, from aforementioned discussion, little atten-tion has been paid to the ISS of stochastic impulsive switchedtime-delay systems, which considerably limits the effectivenessof vector Lyapunov functions.
In this paper, we study the ISS of stochastic impulsiveswitched time-delay systems using the vector Lyapunov func-tion approach. For both the stable continuous dynamics caseand the stable discrete dynamics case, the vector Lyapunov-function-based stability conditions are established to guaranteethe ISS of stochastic impulsive switched time-delay systems.Furthermore, we also study the relationships among the vectorLyapunov function approach, the approach based on the com-parison principle and the scalar Lyapunov function approach. Asa result, according to the vector Lyapunov functions, both thecomparison-principle-based stability conditions and the scalar-Lyapunov-function-based stability conditions are also derived.Compared with the existing works like [5], [33]–[35], and [38]–[41], the main contributions of this paper are threefold.
1) We first propose the stability conditions based on vectorLyapunov functions. In the derived conditions, thecomponents of the vector Lyapunov function are allowedto be coupled, and the information in two consecutive im-pulsive switching intervals is also allowed to be coupled.The coupling items are inevitably encountered when theLyapunov function is used to study time-delay systems[20]. To obtain the stability results, the coupling items areavoided or magnified in the previous works like [42] viasome elementary inequalities like the Cauchy–Schwarzinequality and Young inequality, which results in muchconservatism in the obtained conditions. In the previousworks like [33]–[35], the components of the vector
Lyapunov function were separated, so were the time-delay items and the delay-free items in [4], [5], [12],and[15]. In this paper, the coupling items are introducedin the derived conditions, thereby avoiding the magnifi-cation on the corresponding coupling items, and leadingour results to be more general and less conservative; seenumerical examples in Section V. A special case of ourresults was already used in [43]. Furthermore, for thestable discrete dynamics case, the positive effects of thetime delays on stability analysis are also studied; seealso Theorem 4.
2) According to the vector Lyapunov functions, the stabilityconditions based on the comparison principle are derived,which implies the relationships between the vector Lya-punov function approach and the approach based on thecomparison principle. According to the vector-Lyapunov-function-based stability conditions, the comparison sys-tems are constructed explicitly. However, in the previousworks like [34], [35], [38], and[40], the time delays arenot studied and the comparison systems are assumed tobe existent instead of constructed specifically.
3) The relationships between the vector Lyapunov func-tion approach and the scalar Lyapunov function approachare studied in this paper. Using the summing technique,the vector Lyapunov functions are transformed into thescalar Lyapunov functions. Therefore, the vector Lya-punov function approach and the scalar Lyapunov func-tion approach are connected. Furthermore, according tothe existing results in [5],[12], and[13], based on scalarLyapunov functions, the corresponding stability condi-tions are obtained. In addition, we compare the vectorLyapunov function approach and the scalar Lyapunovapproach via the obtained average dwell-time conditions;see Remarks 9 and 10 and Section V.
As a result, the obtained results in this paper expand greatlythe effectiveness of the vector Lyapunov function approach,and could be applied to a large class of dynamic systems liketime-delay systems, neural systems, and chaotic systems; seeSection V. That is, this paper improves those results in [20],[21], and [34].
The rest of this paper is organized as follows. In section II,the considered problem is formulated and some necessary pre-liminaries are given. The main results of this paper, presentedin Sections III and IV, give different types of stability condi-tions in terms of vector Lyapunov functions. The case that thecontinuous dynamics is stable is studied in Section III, and thecase that the discrete dynamics is stable is studied in Section IV.Two numerical examples are presented in Section V to illustratethe developed results. Conclusions and future works are givenin Section VI.
II. PRELIMINARIES
Let R := (−∞,+∞); R≥t := [t,+∞) and R>t := (t,+∞)for a given t ≥ 0. N := {0, 1, . . .}; N>0 := {1, 2, . . .}. Rn
denotes the n-dimensional Euclidean space. Given two vec-tor x, y ∈ Rn , x � y (x � y) if xi ≥ yi (xi > yi) for alli ∈ {1, . . . , n}. For a given vector or matrix P , P� denotes
656 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 64, NO. 2, FEBRUARY 2019
its transpose. For a matrix P ∈ Rn×n , tr[P ] denotes the traceof P . E denotes the vector all whose components are 1; I de-notes the identity matrix; diag(·) denotes the diagonal matrix.(x, y) := (x�, y�)� for simplicity. Recall from [44, ch. 2.5] thata real matrix P ∈ Rn×n is called a Z-matrix if its off-diagonalentries of P are nonpositive. A Z-matrix P is an M -matrix if allits eigenvalues have positive real parts. Furthermore, it followsfrom [44, Th. 2.2.3] that P is an M -matrix if and only if Pis a Z-matrix and there is a positive vector x ∈ Rn such thatPx � 0. Denote ΩM (P ) := {x ∈ Rn : Px � 0, x � 0}.
In addition, | · | represents the Euclidean vector norm; P{·}denotes the probability measure; E[·] denotes the mathemati-cal expectation. Let PC([a, b]; Rn ) denote the class of piece-wise continuous functions mapping [a, b] to Rn and havingfinite right-hand continuous jumps on [a, b]. For a given func-tion f : R≥t0 → Rn and the initial time t0 ≥ τ > 0, ‖f‖τ :=supt∈[t0 −τ ,t0 ] |f(t)|; ‖f‖[t0 ,t) := supt∈[t0 ,t] |f(t)|; ‖f‖ denotesthe supremum norm on [t0 ,∞). C1,2 stands for the class of thenonnegative functions on R≥t0 × Rn × N>0 , which are con-tinuously differentiable on the first augment and continuouslytwice differentiable on the second augment. Given a functionf : R → R, denote f(t−) := lim sups→0− f(t + s). A functionα : R≥0 → R≥0 is of class K if it is continuous, zero at zero,and strictly increasing; α(t) is of class K∞ if it is of class K andunbounded; α(t) is of class VK (or VK∞) if it is of class K (orK∞) and convex; α(t) is of class CK (or CK∞) if it is of classK (or K∞) and concave. A function β : R≥0 × R≥0 → R≥0 isof class KL if β(s, t) is of class K for each fixed t ≥ 0 anddecreases to zero as t → ∞ for each fixed s ≥ 0. Denote byα−1 the inverse of the function α : R → R.
Consider the following stochastic impulsive switched time-delay system:
t ∈ R≥t0 \ Tx(t) = hσ (t)(x(t−), u(t−)), t ∈ Tx(t) = ξ(t), t ∈ [t0 − τ, t0 ]
(1)
where x(t) ∈ Rnx is the system state, u(t) ∈ PC([t0 ,∞); Rnu )is the external input, and B(t) ∈ Rnw is an Ft-adapted Brow-nian motion defined on a complete probability space (Ω,F,P, {Ft}t≥t0 ). Denote xt := x(t − τ(t)), where the time de-lay τ(t) is bounded with a constant τ > 0. T := {t0 , t1 , . . .}is a given impulsive switching time sequence. The func-tion σ : R≥t0 → M =: {1, . . . , M} is the switching signal,which is piecewise right continuous. The initial functionξ ∈ PC([t0 − τ, t0 ]; Rnx ) is an Ft0 -adapted random variablewith finite E[‖ξ‖2
τ ]. For all i ∈ M, the functions fi : R≥t0 ×Rnx × Rnu → Rnx , gi : R≥t0 × Rnx × Rnu → Rnx ×nw andhi : Rnx × Rnu → Rnx are assumed to be Lipschitz and Borelmeasurable. Suppose that fi(t, 0, 0) ≡ 0, gi(t, 0, 0) ≡ 0 andhi(0, 0) ≡ 0 for all t ∈ R≥t0 . That is, x(t) ≡ 0 is a trivial so-lution of the system (1). It follows from [5], [45], and [46] thatthe system (1) has a unique solution process for all the time.
Remark 1: In this paper, the switching laws for the contin-uous dynamics and the discrete dynamics are assumed to bethe same. If not, then the switching laws for the continuousdynamics and the discrete dynamics could be augmented as a2-D switching law and the following stability analysis could beproceeded along the similar fashion. Similar scenarios could befound in the previous works like [11] and [12]. �
Definition 1 (see[5] and [14]): Given an impulsive switch-ing time sequence T , the system (1) is stochastically input-to-state stable (SISS), if for an arbitrary ε ∈ (0, 1), there existβ ∈ KL, γ ∈ K∞ such that for all ξ ∈ PC([t0 − τ, t0 ]; Rnx ),u ∈ PC([t0 ,∞); Rnu ) and t ∈ R≥t0 ,
P {|x(t)| ≤ β(‖ξ‖τ , t − t0) + γ(‖u‖)} ≥ 1 − ε. (2)
To investigate the SISS property of the system (1), the averagedwell-time condition and the vector Lyapunov functions whosecomponents belong to C1,2 are involved. In the following, theaverage dwell-time and the infinitesimal operator of the C1,2
Lyapunov functions are introduced.Definition 2 (see[5] and [8]): For a switching signal σ(t)
and any t2 > t1 > t0 , let Nσ (t2 , t1) be the switching number ofσ(t) over the interval [t1 , t2). If there exist N0 ≥ 1 and τa > 0such that
t2 − t1τa
− N0 ≤ Nσ (t2 , t1) ≤t2 − t1
τa+ N0 (3)
then N0 and τa are called the chatter bound and the averagedwell time (ADT), respectively.
Definition 3 (see[45]): Given any C1,2 function Vl : R≥0 ×Rnx ×M → R≥0 , where l ∈ L := {1, . . . , L} and L ≤ nx , theinfinitesimal operator of Vl(t, x, σ), σ ∈ M, associated with thecontinuous dynamics of the system (1), is defined as
LVl(t, xt , σ) =∂Vl(t, x, σ)
∂t+
∂Vl(t, x, σ)∂x
fσ (t, xt , u)
+12tr
[
g�σ (t, xt , u)∂2Vl(t, x, σ)
∂x2 gσ (t, xt , u)]
.
In Definition 3, the vector Lyapunov function is defined as
According to the previous works [33]–[35], [39], it is assumedthat the dimension of the vector Lyapunov function is not morethan that of the system state. Especially, if L ≡ 1, then the vectorLyapunov function is reduced to be the scalar one as in [8], [14],and [23].
Remark 2: For each l ∈ L, Vl(t, x, σ) is in fact a multipleLyapunov function due to the existence of σ. That is, the ap-plied vector Lyapunov function is a vector version of the mul-tiple Lyapunov functions. In addition, it follows from [29, ch.2.6] that the multiple vector Lyapunov function could be trans-formed further into matrix Lyapunov function; see [29] for moredetails. �
It follows from the Ito’s differential formula in [45, ch. 1] thatfor all l ∈ L, the derivative of Vl(t, x, σ) is given by
dVl(t, x, σ) = LVl(t, xt , σ)dt +∂Vl(t, x, σ)
∂x
× gσ (t)(t, xt , u)dB(t).
Moreover, dE[Vl(t, x, σ)] = E[LVl(t, xt , σ)]dt. Given a func-tion W : R≥0 → R≥0 , the upper Dini derivative of W (t) isdefined as D+W (t) := lim sups→0+
W (t+s)−W (t)s . In addi-
tion, if E[LVl(t, xt , σ)] is continuous, then it follows from[5] and [13] that E[LVl(t, xt , σ)] = D+W (t), where W (t) =E[Vl(t, x(t), σ(t))].
Before presenting the main results, a vector version of thecomparison principle is given for impulsive time-delay systems.
REN AND XIONG: VECTOR-LYAPUNOV-FUNCTION-BASED ISS OF STOCHASTIC IMPULSIVE SWITCHED TIME-DELAY SYSTEMS 657
Lemma 1: Assume that X,Y ∈ PC([t0 − τ,∞); Rn ) andthat Ψ1 ,Ψ2 ∈ PC([t0 ,∞); Rn ) are continuous in [tk , tk+1),where k ∈ N. Suppose there exist continuously nondecreasingfunctions Φ1 : Rn → Rn , Φ2 : Rn → Rn and a continuouslyincreasing function Υ : Rn → Rn such that for all k ∈ N,
⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
D+X(t) � Φ1(X(t)) + sup−τ≤s≤0 Φ2(X(t + s))
+Ψ1(|u(t)|), t ∈ (tk , tk+1) (a)
X(tk ) � Υ(X(t−k )) + Ψ2(t−k ) (b)
X(t) � Π(t), t ∈ [t0 − τ, t0) (c)
(4)
and⎧⎪⎪⎪⎪⎨
⎪⎪⎪⎪⎩
D+Y(t) � Φ1(Y(t)) + sup−τ≤s≤0 Φ2(Y(t + s))
+Ψ1(|u(t)|), t ∈ (tk , tk+1) (a)
Y(tk ) � Υ(Y(t−k )) + Ψ2(t−k ) (b)
Y(t) � Π(t), t ∈ [t0 − τ, t0). (c)
(5)
Then, X(t) � Y(t) for all t ∈ [t0 ,∞).Proof: To prove Lemma 1, we first prove that X(t) � Y(t)
for t ∈ [t0 , t1) via reductio ad absurdum. Second, using themathematical induction, we prove that X(t) � Y(t) for t ∈[t0 ,∞).
Because X(t) � Π(t) � Y(t) holds for all t ∈ [t0 − τ, t0),Υ(t) is increasing and the domain of Ψ2(t) is [t0 ,∞), it fol-lows from (b) in (4) and (b) in (5) that Y(t0) � Υ(Y(t−0 )) �Υ(X(t−0 )) � X(t0), which implies that X(t) � Y(t) at t = t0 .
Suppose that X(t) � Y(t) holds for t ∈ (t0 , t1). If not, thenthere exists a t ∈ (t0 , t1) such that X(t) � Y(t) does not hold.In addition, there exists at least a i ∈ {1, . . . , n} such that the ithcomponent of X(t), i.e., Xi(t), satisfies Xi(t) > Yi(t). Definet := inf{t ∈ (t0 , t1)|Xi(t) > Yi(t), i ∈ {1, . . . , n}} and j :=min{i ∈ {1, . . . , n}|Xi(t) ≥ Yi(t)}. Therefore, we have that
X(t) � Y(t), t ∈ (t0 , t) (6)
Xj (t) = Yj (t) (7)
Xj (t) > Yj (t), t ∈ (t, t + Δt) (8)
where Δt > 0 is arbitrarily small. Thus, it obtains from (7) and(8) that D+Xj (t) ≥ D+Yj (t). On the other hand, it followsfrom the definition of t and (6) that X(t − τ(t)) � Y(t − τ(t)).Hence, combining (a) in (4) and (a) in (5) yields that D+X(t) ≺D+Y(t), which implies that D+Xj (t) < D+Yj (t). This isa contradiction. Therefore, there do not exist j ∈ {1, . . . , n}and t ∈ (t0 , t1) such that Xj (t) > Yj (t), and it concludes thatX(t) � Y(t) for all t ∈ (t0 , t1).
In the following, suppose that X(t) � Y(t) holds for all t ∈[t0 , tk ), k ∈ N>0 . As a result, it follows that X(t) � Y(t) fort ∈ [tk − τ, tk ). Because of (b) in (4) and (b) in (5), one has that
X(tk ) � Υ(X(t−k )) + Ψ2(t−k )
� Υ(Y(t−k )) + Ψ2(t−k ) � Y(tk )
that is, X(tk ) � Y(tk ).If X(t) � Y(t) is not valid for some t ∈ (tk , tk+1), k ∈ N>0 ,
then there exist t ∈ (tk , tk+1) and j ∈ {1, . . . , n} such thatXj (t) > Yj (t). Along the same line as the proof for the case of(t0 , t1), we have that X(t) � Y(t) for all t ∈ [tk , tk+1).
Based to the mathematical induction, it follows that X(t) �Y(t) for all t ∈ [t0 ,∞). Therefore, the proof is completed. �
Remark 3: Lemma 1 generalizes the classic comparisonprinciple in [47, Sec. 3.4] and the scalar version of compari-son principle in [5, Lemma 1]. Lemma 1 will be used in thefollowing sections to bridge the connection between the ob-tained results and the results in previous works [34], [38] basedon the comparison principle. �
III. SISS WITH VECTOR LYAPUNOV FUNCTIONS AND ADT
In this section, we study the SISS of the system (1) with thecase that the continuous dynamics is stable. Using the vectorLyapunov functions and the ADT condition, sufficient condi-tions are established for the SISS of the system (1), and the re-lationships between the obtained results and the previous worksare also discussed.
Theorem 1: Consider the system (1). Assume that there ex-ist locally Lipschitz Lyapunov functions Vl : R≥0 × Rnx ×M → R≥0 , l ∈ L, σ ∈ M, α1 ∈ VK∞, α2 ∈ CK∞, and ϕl ∈K∞, positive matrices λ := diag(λ1 , . . . , λL ) and μ :=diag(μ1 , . . . , μL ), and nonnegative matrices P := [pij ]L×L ,Q := [qij ]L×L , and R := [rij ]L×L such that μ − I ≥ 0, λ −P − Q is a nonsingular M -matrix and
A.1) for all l ∈ L and all t ∈ R≥t0 , α1(|x(t)|) ≤ V l(t) ≤α2(|x(t)|);
A.2) for all l ∈ L, σ ∈ M and all t ∈ R≥t0 \T , V l(t) ≥ϕl(|u(t)|) implies
LVl(t, xt , σ) ≤ −λlV l(t) +L∑
j=1
plj
√
V l(t)V j (t)
+L∑
j=1
qlj sup−τ≤s≤0
√
V l(t)V j (t + s)
A.3) for all l ∈ L and all k ∈ N>0 , V l(t) ≥ ϕl(|u(t)|) im-plies
V l(tk ) ≤ μlV l(t−k ) +L∑
j=1
rlj
√
V l(t−k )V j (t−k )
A.4) τa > maxl∈Lln(μl +
∑ Lj = 1 rl j )
η , where η ∈ (0, η), η :=
sup{θ > 0 : 2 ζl θ − ζlλl +∑L
j=1 plj ζj +∑L
j=1 qlj ζj eθ τ
< 0, l ∈ L} with ζ ∈ ΩM (λ − P − Q) and maxl∈L
{ζl} ≥ 1where V l(t) := Vl(t, x(t), σ(t)), then the system (1) is SISS.Proof: The proof is partitioned into the following three parts.
First, we analyze the existence of η. Second, the boundednessof the Lyapunov functions V l(t) is established via reductioad absurdum. Finally, based on the bounds of the Lyapunovfunctions and the ADT condition [i.e., (A.4)], the convergenceof the system state is obtained, which in turn guarantees theSISS of the system (1).
Part 1: Define Γl(θ) := 2ζlθ − ζlλl +∑L
j=1 plj ζj +∑L
j=1
qlj ζj eθτ , where ζ ∈ ΩM (λ − P − Q) and maxl∈L{ζl}≥1.
Since λ − P − Q is nonnegative M -matrix and Q is non-negative matrix, it is obvious that Γl(0) = −ζlλl +
∑Lj=1
plj ζj +∑L
j=1 qlj ζj < 0 and that Γl(θ) → ∞ as θ → ∞. In ad-
dition, Γ′l(θ) := 2ζl +
∑Lj=1 qlj ζj θe
θτ > 0. Thus, there existsa unique θl > 0 such that Γl(θl) = 0, and Γl(θ) < 0 for allθ ∈ (0, θl). It also implies that η := minl∈L{θl}.
658 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 64, NO. 2, FEBRUARY 2019
Part 2: Since α1 ∈ VK∞ and α2 ∈ CK∞, it obtains from(A.1) and Jensen’s inequality in [48, ch. II, 18.3] that for alll ∈ L and all t ∈ R≥t0 ,
α1(E[|x(t)|]) ≤ E[V l(t)] ≤ α2(E[|x(t)|]). (9)
Thus, it follows that E[V l(t0)] ≤ α2(E[‖ξ‖τ ]).Taking expectation and using Holder inequality in [48, ch. II,
10.2], it obtains from (A.2)–(A.3) that, if V l(t) ≥ ϕl(|u(t)|)for all t ∈ R≥t0 , then for all l ∈ L, σ ∈ M and all t ∈ R≥t0 \T ,
E[LVl(t, xt , σ)]≤−λl E[V l(t)]+L∑
j=1
plj
√
E[V l(t)]E[V j (t)]
+L∑
j=1
qlj sup−τ≤s≤0
√
E[V l(t)]E[V j (t + s)]
(10)
and that for all l ∈ L and all k ∈ N>0 ,
E[V l(tk )] ≤ μl E[V l(t−k )] +L∑
j=1
rlj
√
E[V l(t−k )]E[V j (t−k )].
(11)
In the following, consider the following two cases: the firstcase that V l(t) ≥ ϕl(|u(t)|) for all t ∈ R≥t0 and the secondcase that V l(t) < ϕl(|u(t)|) for all t ∈ R≥t0 . For the first case,we prove that for all l ∈ L and any η ∈ (0, η),
j=1 rlj},and ζ := maxl∈L{ζl}. From the assumption that maxl∈L{ζl} ≥1, it follows that (12) holds at t0 .
To this end, define U l(t) :=√
2V l(t) for all l ∈ L, then itobtains that
D+ E[V l(t)] := E[U l(t)]D+ E[U l(t)]. (13)
Furthermore, for all l ∈ L, (10)–(11) are equivalent to the fol-lowing: for all l ∈ L and t ∈ R≥t0 \T ,
D+ E[U l(t)] ≤ −λl
2E[U l(t)] +
L∑
j=1
plj
2E[U j (t)]
+L∑
j=1
qlj
2sup
−τ≤s≤0E[U j (t + s)] (14)
and for all l ∈ L and all k ∈ N>0 ,
E[U l(tk )] ≤ μl E[U l(t−k )] +L∑
j=1
rlj E[U j (t−k )]. (15)
Accordingly, (12) equals to
E[U l(t)] ≤ maxl∈L
{√2μN (t,t0 )ζle
−η (t−t0 ) E[V 0(t0)]1/2}
=: maxl∈L
{Ul(t − t0)}. (16)
If (12) is not valid, then there are following two scenarios: (12)does not hold in certain impulsive switching interval and (12)does not hold at certain impulsive switching time. For the first
scenario, we prove that E[U l(t)] ≤ Ul(t − t0) for all l ∈ L andall t ∈ R≥t0 , which implies that (16) [i.e., (12)] is valid. If not,suppose thatE[U l(t)] ≤ Ul(t − t0) holds for all [t0 , tk ] and that(tk , tk+1) is the first interval such that E[U l(t)] ≤ Ul(t − t0)does not hold for l = � ∈ L. Define t∗ := inf{t ∈ (tk , tk+1) :E[U �(t)] > U�(t − t0)} such that
E[U �(t∗)] = U�(t∗ − t0) (17)
E[U �(t)] > U�(t − t0), t ∈ (t∗, t∗ + Δt) (18)
where Δt > 0 is arbitrarily small. It obtains from (17) and (18)that
D+ E[U �(t∗)] ≥ D+U�(t∗ − t0). (19)
However, it follows from the definition of η and (17)–(18) that
D+U�(t∗ − t0) >
⎛
⎝−λ�
2ζ� +
L∑
j=1
p�j
2ζj +
L∑
j=1
q�j
2ζj e
ητ
⎞
⎠
×√
2μN (t∗,t0 )e−η (t∗−t0 ) E[V 0(t0)]1/2
≥ −λ�
2E[U �(t∗)] +
L∑
j=1
p�j
2E[U �(t∗)]
+L∑
j=1
q�j
2sup
−τ≤s≤0eη (τ +s)μN (t∗,t∗+s)
× E[U �(t∗ + s)]
≥ −λ�
2E[U l(t∗)] +
L∑
j=1
p�j
2E[U j (t∗)]
+L∑
j=1
q�j
2sup
−τ≤s≤0E[U j (t∗ + s)]
≥ D+ E[U �(t∗)]
which contradicts with (19). As a result, E[U l(t)] ≤ Ul(t − t0)for all l ∈ L and all t ∈ (tk , tk+1). It implies that (16) [i.e., (12)]holds for all t ∈ (tk , tk+1).
For the second scenario, assume that (16) [i.e., (12)] holdsfor all t ∈ [t0 , tk ) but does not hold at tk , k ∈ N>0 . That is,E[U �(tk )] > maxl∈L{Ul(tk − t0)} for some � ∈ L. However,it obtains from (15) that at tk ,
E[U �(tk )] ≤ μl E[U l(t−k )] +L∑
j=1
rlj E[U j (t−k )]
≤
⎛
⎝μ� +L∑
j=1
r�j
⎞
⎠√
2μN (t−k ,t0 )ζle−η (t−k −t0 )
× E[V 0(t0)]1/2
≤√
2μN (tk ,t0 ) ζe−η (tk −t0 ) E[V 0(t0)]1/2
= maxl∈L
{Ul(tk − t0)}
which is a contradiction. That is, (16) [i.e., (12)] holds for t = tk .Based on the aforementioned analysis and using mathe-
matical induction, it concludes that if V l(t) ≥ ϕl(|u(t)|) for
REN AND XIONG: VECTOR-LYAPUNOV-FUNCTION-BASED ISS OF STOCHASTIC IMPULSIVE SWITCHED TIME-DELAY SYSTEMS 659
all t ∈ R≥t0 , then (12) is valid for all t ∈ R≥t0 . The secondcase is V l(t) < ϕl(|u(t)|) for all t ∈ R≥t0 , which implies thatE[V l(t)] < ϕl(|u(t)|) for all t ∈ R≥t0 . Combining the first caseand the second case, we obtain that for all l ∈ L and all t ∈ R≥t0 ,
where β(v, t) := ε−1e−2Λt μ2N0 ζ2Lα2(v). As a result, it fol-lows from (A.1) that for all t ∈ R≥t0 ,
P {|x(t)| ≤ β(E[‖ξ‖τ ], t − t0) + γ(‖u‖)} ≥ 1 − ε
where β(v, t) := α−11 (2β(v, t)), γ(v) := α−1
1 (2ϕ(v)), andϕ(v) := maxl∈L ε−1ϕl(v). Thus, the SISS of the system (1)is established and the proof is completed. �
Remark 4: Let us examine the conditions (A.2) and (A.3).In (A.2) and (A.3), different components of the vector Lya-punov function are allowed to be coupled, so are the time-delay items and the delay-free items. Since the vector Lyapunovfunction is applied to stability analysis of stochastic impul-sive switched time-delay systems, the coupling items are in-evitably encountered; see [20]. However, to obtain the stabilityresults in the previous works, there are two methods to dealwith the coupling items. The first method is to separate dif-ferent items. For instance, the time-delay items and the delay-free items are separated in [4], [5], [12], and[15], and differentcomponents of the vector Lyapunov functions are separatedin [33]–[35]. The second method is to use many techniquesand elementary inequalities, such as Cauchy–Schwarz inequal-ity and Young inequality, to deal with the coupling items; see[42] and references therein. Through such two methods, theobtained stability conditions have much more conservatism. InTheorem 1, the coupling items are introduced in the condi-tions (A.2)–(A.3). As a result, our result is more general andless conservative, and the magnification on the correspondingcoupling item is avoided. On the other hand, the condition(A.2) is different from the conditions in the previous works[5], [12] based on the scalar Lyapunov functions. In (A.2), thederivatives of the Lyapunov functions in an impulsive switch-ing interval depends on the information in both the currentimpulsive switching interval and the last impulsive switchinginterval. Since V l(t + s) = Vl(t + s, x(t + s), σ(t + s)) ands ∈ [−τ, 0], σ(t + s) does not necessarily equal to σ(t). If
σ(t + s) �= σ(t), then the condition (A.2) contains the infor-mation in the last impulsive switching interval, which impliesthat the Lyapunov functions for different subsystems could alsobe coupled in this study. However, the derivatives of the Lya-punov functions in an impulsive switching interval in [5] and[12] depend only on the information in the current impulsiveswitching interval. �
Remark 5: Theorem 1 is general and recovers the stabilityresults in [20] for stochastic delayed differential systems and in[21] for delayed neural networks as the special cases. Moreover,ζ � E is required in [20] and [21], but relaxed in this study. Inaddition, if there is no external input, then Theorem 1 impliesthe stochastic global asymptotic stability of the system (1); see[43]. If L = 1, that is, the vector Lyapunov function is reducedto be the scalar one, then Theorem 1 is similar to the results in[5], [11], and [12]. �
Remark 6: The conditions in Theorem 1 guarantee the SISSof the system (1) in the case that the continuous dynamicsis stable. If both the continuous dynamics and the discretedynamics are stable, then Theorem 1 is also valid. In thiscase, (μ + R)E < E and it follows from (A.4) that there areno constraints on τa . In addition, the ADT condition (A.3)depends on the choice of ζ. If ζ = θE for certain θ ≥ 1,then η := sup{θ > 0 : 2θ − λl +
∑Lj=1 plj +
∑Lj=1 qlj e
θτ <
0, l ∈ L}, which does not depend on the choice of ζ. �Based on the proof of Theorem 1, the following corollary is
an immediate consequence and provides alternative sufficientconditions for the SISS of the system (1).
Corollary 1: Consider the system (1). Assume that thereexist locally Lipschitz Lyapunov functions Ul : R≥0 × Rnx ×M → R≥0 , l ∈ L, σ ∈ M, α1 ∈ VK∞, α2 ∈ CK∞, andϕl, ϕl ∈ K∞, positive matrices λ := diag(λ1 , . . . , λL ) and μ :=diag(μ1 , . . . , μL ), and nonnegative matrices P := [pij ]L×L ,Q := [qij ]L×L , and R := [rij ]L×L such that μ − I ≥ 0, λ −P − Q is a nonsingular M -matrix, the conditions (A.1) and(A.4) hold, and
B.1) for all l ∈ L, σ ∈ M and all t ∈ R≥t0 \T , U l(t) ≥ϕl(|u(t)|) implies
2LUl(t, xt , σ) ≤ −λlU l(t) +L∑
j=1
pljU j (t)
+L∑
j=1
qlj sup−τ≤s≤0
U j (t + s)
B.2) for all l ∈ L and all k ∈ N>0 , U l(t) ≥ ϕl(|u(t)|) im-plies
U l(tk ) ≤ μlU l(t−k ) +L∑
j=1
rljU j (t−k )
then the system (1) is SISS.
A. Stability Analysis via Comparison Principle
In this subsection, a comparison system is constructed firstbased on the stability conditions in Theorem 1 and Corollary 1.According to the constructed comparison system, we proposethe stability conditions via the comparison principle.
To begin with, let us examine the relationship between (A.2)–(A.3) and (B.1)–(B.2). Taking expectation, it obtains from (B.1)
660 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 64, NO. 2, FEBRUARY 2019
and (B.2) that if U l(t) ≥ ϕl(|u(t)|) for all t ∈ R≥t0 , then forall l ∈ L, σ ∈ M and all t ∈ R≥t0 \T ,
2E[LUl(t, xt , σ)] ≤ −λl E[U l(t)] +L∑
j=1
plj E[U j (t)]
+L∑
j=1
qlj sup−τ≤s≤0
E[U j (t + s)] (23)
and for all l ∈ L and all k ∈ N>0 ,
E[U l(tk )] ≤ μl E[U l(t−k )] +L∑
j=1
rlj E[U j (t−k )]. (24)
Therefore, (23)–(24) and (14)–(15) are equivalent. The differ-ence between (A.2)–(A.3) and (B.1)–(B.2) lies in that thereare coupling items in (A.2)–(A.3) instead of (B.1)–(B.2). Es-pecially, if there are no random disturbances, then the condi-tions in Theorem 1 and Corollary 1 are equivalent by settingU l(t) :=
√V l(t).
Furthermore, using [27, Proposition 2.6], there exists ϕl ∈K∞ (which may depends on ϕl) such that the conditions (B.1)–(B.2) could be rewritten as the following vector form:
2LU(t, xt , σ) � −(λ − P )U(t) + Q sup−τ≤s≤0
U(t + s)
+ ϕ(|u(t)|), t ∈ R≥t0 \T (25)
U(t) � (μ + R)U(t−) + ϕ(|u(t−)|), t ∈ T (26)
where U(t) := U(t, x(t), σ(t)), and ϕ(v) := (ϕ1(v), . . . ,ϕL (v)). Define the comparison system P with the followingform:
⎧⎪⎨
⎪⎩
2D+y(t) = −(λ − P )y(t) + Q sup−τ≤s≤0 y(t + s)
+ϕ(|u(t)|) + δ, t ∈ R≥t0 \Ty(t) = (μ + R)y(t−) + ϕ(|u(t−)|), t ∈ T
where y(t) ∈ RL is the system state, and δ > 0 is arbitrary.Based on Lemma 1, the following proposition is obtained.
Proposition 1: Consider the system (1). Assume that thereexist a locally Lipschitz Lyapunov function U : R≥0 × Rnx ×M → RL
≥0 , α1 ∈ VK∞, α2 ∈ CK∞, ϕl ∈ K∞, positive ma-trices λ := diag(λ1 , . . . , λL ) and μ := diag(μ1 , . . . , μL ), andnonnegative matrices P := [pij ]L×L , Q := [qij ]L×L , and R :=[rij ]L×L such that μ − I ≥ 0, λ − P − Q is a nonsingular M -matrix, the condition (A.1) and the inequalities (25)–(26) hold.Suppose that the comparison system P has a unique solution.If U(t0) � y(t0), then the SISS of the comparison system Pimplies the SISS of the system (1).
Proof: Since U(t0) � y(t0), it follows that E[U(t0)] �y(t0). Using Lemma 1, it obtains that E[U(t)] � y(t) for allt ∈ R≥t0 . In the sequel, according to Markov’s inequality and(A.1), it obtains that for an arbitrary ε ∈ (0, 1),
P
{
|x(t)| ≤ maxl∈L
{α−11 (ε−1 |yl(t)|)}
}
≥ 1 − ε ∀t ∈ R≥t0 .
As a result, the SISS property of the comparison system Pimplies that the SISS property of the system (1). �
Remark 7: Proposition 1 casts a new light on the stabilityconditions in Corollary 1 (or Theorem 1), and bridges the con-nection between the obtained conditions in Corollary 1 (orTheorem 1) and those in the previous works [34], [35], [38],[40] using the comparison principle. The key differences are asfollows: 1) the time delays are considered in this study insteadof in the previous works [34], [35], [40]; and 2) the compari-son systems are assumed to be existent in the previous works,whereas the comparison system is constructed in Proposition 1.In addition, the ADT condition is omitted in Proposition 1. Thereason is that the ADT condition in Theorem 1 is to bring theLyapunov functions to decrease along the time line, which isnot required for the comparison system. �
B. Stability Analysis via Scalar Lyapunov Functions
In what follows, the relationship between the vector Lyapunovfunction approach and the scalar Lyapunov function approachare shown. By summing all the components of the vector Lya-punov function, we transform the vector Lyapunov function intothe scalar one. In this way, the stability conditions are derived inthe following theorem. Similar to Proposition 1, the followingtheorem builds up a connection between Theorem 1 and theresults in the previous works [5], [12], [14] based on the scalarLyapunov functions.
Theorem 2: Consider the system (1). Assume that there ex-ist locally Lipschitz Lyapunov functions Vl : R≥0 × Rnx ×M → R≥0 , l ∈ L, σ ∈ M, α1 ∈ VK∞, α2 ∈ CK∞, and ϕl ∈K∞, positive matrices λ := diag(λ1 , . . . , λL ) and μ :=diag(μ1 , . . . , μL ), and nonnegative matrices P := [pij ]L×L ,Q := [qij ]L×L , and R := [rij ]L×L such that μ − I ≥ 0, 2λ −(P + P� + Q + Q�) > 0, the conditions (A.1)–(A.3) hold,and
τa >ln μ
�0, �0 ∈ (0,�) (27)
where μ := maxl∈L{μl + 12
∑Lj=1(rlj + rjl)}, � := maxl∈L
{�l} and �l is the solution to the equation
−λl +12
L∑
j=1
(plj + pjl + qlj ) +12
L∑
=1
qjleθτ + θ = 0 (28)
then the system (1) is SISS.Proof: Analogous to the proof of Theorem 1 and using [27,
Proposition 2.6] and the arithmetic mean-geometric mean (AM-GM) inequality, it follows from the conditions (A.2)–(A.3) thatthere exists ϕl ∈ K∞ such that, for all l ∈ L, σ ∈ M and allt ∈ R≥t0 \T ,
E[LVl(t, xt , σ)] ≤ −λl E[V l(t)]
+L∑
j=1
plj
√
E[V l(t)]E[V j (t)]
+L∑
j=1
qlj sup−τ≤s≤0
√
E[V l(t)]E[V j (t + s)] + ϕl(|u(t)|)
REN AND XIONG: VECTOR-LYAPUNOV-FUNCTION-BASED ISS OF STOCHASTIC IMPULSIVE SWITCHED TIME-DELAY SYSTEMS 661
≤
⎛
⎝−λl +12
L∑
j=1
plj +12
L∑
j=1
qlj
⎞
⎠E[V l(t)]
+12
L∑
j=1
plj E[V j (t)]
+12
L∑
j=1
qlj sup−τ≤s≤0
E[V j (t + s)] + ϕl(|u(t)|) (29)
and that for all l ∈ L and all k ∈ N>0 ,
E[V l(tk )] ≤ μl E[V l(t−k )] +L∑
j=1
rlj
√
E[V l(t−k )]E[V j (t−k )]
+ ϕl(|u(t−k )|)
≤
⎛
⎝μl +12
L∑
j=1
rlj
⎞
⎠E[V l(t−k )] +12
L∑
j=1
rlj E[V j (t−k )]
+ ϕl(|u(t−k )|). (30)
Define V(t, x, σ) :=∑L
l=1 Vl(t, x, σ) and V(t) :=∑L
l=1V l(t). Combining (29) and (30) yields that for all t ∈ R≥t0 \T ,
E[LV(t, xt , σ)] ≤ maxl∈L
⎧⎨
⎩−λl +
12
L∑
j=1
(plj
+ pjl + qlj )
}
E[V(t)]
+12
maxl∈L
⎧⎨
⎩
L∑
j=1
qjl
⎫⎬
⎭sup
−τ≤s≤0E[V(t + s)] + ϕ(|u(t)|)
(31)
and for all k ∈ N>0 ,
E[V(tk )] ≤ maxl∈L
⎧⎨
⎩μl +
12
L∑
j=1
(rlj + rjl)
⎫⎬
⎭E[V(t−k )]
+ ϕ(|u(t−k )|) (32)
where ϕ(v) :=∑L
j=1 ϕl(v).According to (31) and (32) and the ADT condition (27), it
follows from [5, Th. 1] that the system (1) is SISS. Therefore,the proof is completed. �
Remark 8: The proofing strategy of Theorem 2 is based onthe decoupling technique and the result in [5] using scalar Lya-punov functions. All the coupled items in (A.2) and (A.3) aredecoupled first by the AM-GM inequality. Then, by summing allthe components of the vector Lyapunov function, the methodsin [5] could be implemented. As a result, Theorem 2 providesa connection between the vector Lyapunov function approachand the scalar Lyapunov function approach. In the same way, wecould derive the similar result from the conditions in Corollary 1more directly. In addition, after transforming the conditions(A.2) and (A.3) into (31) and (32), some other stability prop-erties could be obtained for the system (1), such as stochasticintegral input-to-state stability (SiISS) and weighted ISS; see
[5] for more details. On the other hand, if the scalar Lyapunovfunction is not the sum of all the components of the vector Lya-punov function but satisfies (31) and (32), then the obtainedstability results are not related to Theorem 1, and may havedifferent effects on the system performances; see Example 2 inSection V. Even some dynamical systems cannot be studied viathe scalar Lyapunov functions; see Section IV in [21]. �
Remark 9: Let us compare the obtained ADT conditions inTheorems 1 and 2. The ADT condition (A.4) is based on thevector Lyapunov function and the ADT condition (27) is basedon the scalar Lyapunov function. In Theorem 1, the ADT condi-tion (A.4) depends on the vector ζ ∈ ΩM (λ − P − Q), whichin turn brings about the conservatism on the ADT (see alsoRemark 6). In Theorem 2, the usage of the AM-GM inequalityleads to the conservatism on the ADT condition (27). There-fore, both the ADT conditions in Theorems 1 and 2 have theirown conservatism. On the other hand, from Theorems 1 and 2,for all l ∈ L, define Γ1l(θ) := ζlθ − 1
2 ζlλl + 12
∑Lj=1 plj ζj +
12
∑Lj=1 qlj ζj e
θτ and Γ2l(θ) := θ − λl + 12
∑Lj=1(plj + qlj +
pjl) + 12
∑Lj=1 qjle
θτ . If P,Q, and R are symmetric and ζ =E, then Γ2l(0) < Γ1l(0) < 0 and Γ′
1l(θ) = Γ′2l(θ) > 0. As a
result, η ≤ �, which implies that the lower bound of the ADTin Theorem 1 is not smaller than that in Theorem 2. However,in other cases, the ADT conditions in Theorems 1 and 2 arenot related and the advantages of such two theorems depend onthe considered systems. For instance, even if P,Q, and R areasymmetric and ζ = E, then the lower bound of the ADT inTheorem 1 may be still larger than that in Theorem 2; seeExample 1 in Section V. �
IV. SISS WITH VECTOR LYAPUNOV FUNCTIONS AND
REVERSE ADT
In this section, we study the case that the discrete dynamics isstable but the continuous dynamics is unstable. Using the vectorLyapunov functions and the reverse ADT condition (see also[23]), different types of stability conditions are derived for thesystem (1).
Theorem 3: Consider the system (1). Assume that thereexist locally Lipschitz Lyapunov functions Vl : R≥0 ×Rnx ×M → R≥0 , l ∈ L, σ ∈ M, α1 ∈ VK∞, α2 ∈ CK∞,and ϕl ∈ K∞, positive matrices λ := diag(λ1 , . . . , λL ) andμ := diag(μ1 , . . . , μL ), and nonnegative matrices P :=[pij ]L×L ,Q := [qij ]L×L , and R := [rij ]L×L such that (λ +P + Q)E � E, (μ + R)E ≺ E, the condition (A.1) holds, and
C.1) for all l ∈ L, σ ∈ M and all t ∈ R≥t0 \T , V l(t) ≥ϕl(|u(t)|) implies
LVl(t, xt , σ) ≤ λlV l(t) +L∑
j=1
plj
√
V l(t)V j (t)
+L∑
j=1
qlj sup−τ≤s≤0
√
V l(t)V j (t + s)
C.2) for all l ∈ L and all k ∈ N>0 , V l(t) ≥ ϕl(|u(t)|) im-plies
V l(tk ) ≤ μlV l(t−k ) +L∑
j=1
rlj
√
V l(t−k )V j (t−k )
662 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 64, NO. 2, FEBRUARY 2019
C.3) τa < ln μ−η , where η < η ≤ η + 1, μ := maxl∈L{μl +
∑Lj=1 rlj} and η := maxl∈L{−2πlω + πlλl +
∑Lj=1
plj πj +∑L
j=1 qlj πj μ−N 0 eτ = 0} with maxl∈L{πl}≥1
then the system (1) is SISS.Proof: Following the similar line as the proof of Theorem 1,
the proof of Theorem 3 is divided into two parts. The first partis to bound the vector Lyapunov function. The second part is toestablish the convergence of the system state according to thebound of the vector Lyapunov function and the reverse ADTcondition [i.e., (C.3)].
Part 1: Similar to the proof of Theorem 1, from (C.1) and(C.2), we consider two cases: V l(t) ≥ ϕl(|u(t)|) and V l(t) ≤ϕl(|u(t)|). For the case V l(t) ≥ ϕl(|u(t)|), it obtains that forall l ∈ L, σ ∈ M and all t ∈ R≥t0 \T ,
E[LVl(t, xt , σ)] ≤ λl E[V l(t)]
+L∑
j=1
pij
√
E[V l(t)]E[V j (t)]
+L∑
j=1
qij sup−τ≤s≤0
√
E[V l(t)]E[V j (t + s)] (33)
and that for all l ∈ L and all k ∈ N>0 ,
E[V l(tk )] ≤ μl E[V l(t−k )] +L∑
j=1
rij
√
E[V l(t−k )]E[V j (t−k )].
(34)
Based on (33) and (34), we prove that for all l ∈ L, and anyη ∈ (η, η + 1],
where π := maxl∈L{πl}. Obviously, (35) holds at t0 . Similarto the proof of Theorem 1, define U l(t) :=
√2V l(t). Conse-
quently, (35) is equivalent to
E[U l(t)] ≤ maxl∈L
{πl}√
2μN (t,t0 )eη (t−t0 ) E[V 0(t0)]1/2
=: maxl∈L
{Al(t − t0)}. (36)
To this end, we consider following two scenarios: (35) failsin certain impulsive switching interval and (35) fails at certainimpulsive switching time. If E[U l(t)] ≤ Al(t − t0) is valid in[t0 , tk ] and invalid in certain (tk , tk+1), k ∈ N>0 , then thereexist � ∈ L and t∗ ∈ (tk , tk+1) such that
E[U �(t∗)] = A�(t∗ − t0) (37)
E[U �(t)] > A�(t∗ − t0), t ∈ (t∗, t∗ + Δt) (38)
where Δt > 0 is arbitrarily small. Therefore, we obtain that
D+ E[U �(t∗)] ≥ D+A�(t∗ − t0). (39)
On the other hand, it follows from (C.3), (33), and (34) that
D+A�(t∗ − t0) =√
2ηπ�μN (t∗,t0 )eη (t∗−t0 ) E[V 0(t0)]1/2
>
⎛
⎝λ�
2π� +
L∑
j=1
p�j
2πj +
L∑
j=1
q�j
2πj μ
−N0 eτ
⎞
⎠
×√
2μN (t∗,t0 )eη (t∗−t0 ) E[V 0(t0)]1/2
≥ λ�
2E[U l(t∗)] +
L∑
j=1
p�j
2E[U j (t∗)]
+L∑
j=1
q�j
2μ−N0 eτ sup
−τ≤s≤0e−ηs μN (t∗,t∗+s) E[U j (t∗ + s)].
Since s ∈ [−τ, 0], it follows from Definition 2 and thereverse ADT condition (C.3) that μ−N0 eτ−ηs μN (t∗,t∗+s) ≥eτ−(η+ln μ/τa )s ≥ 1. As a result, one has
λ�
2E[U l(t∗)] +
L∑
j=1
p�j
2E[U j (t∗)]
+L∑
j=1
q�j
2μ−N0 eτ sup
−τ≤s≤0e−ηs μN (t∗,t∗+s) E[U j (t∗ + s)]
≥ λ�
2E[U l(t∗)] +
L∑
j=1
p�j
2E[U j (t∗)]
+L∑
j=1
q�j
2sup
−τ≤s≤0E[U j (t∗ + s)]
≥ D+ E[U �(t∗)]
which is a contradiction. Therefore, it concludes that (35) isvalid for all t ∈ (tk , tk+1), k ∈ N>0 .
For the second scenario, suppose that (35) holds for all t ∈[t0 , tk ) and fails at tk , k ∈ N>0 . However, for all l ∈ L andt = tk , it follows from (15) and (34) that
E[U l(tk )] ≤ μl E[U l(t−k )] +L∑
j=1
rij E[U j (t−k )]
≤
⎛
⎝μl +L∑
j=1
rlj
⎞
⎠ μN (t−k ,t0 ) ξeη (t−k −t0 ) E[V 0(t0)]1/2
≤ μN (tk ,t0 ) πeη (tk −t0 ) E[V 0(t0)]1/2
which is a contradiction. That is, (35) holds for t = tk , k ∈ N.Therefore, based on the aforementioned analysis and accord-
ing to the mathematical induction, we conclude that for thefirst case, (35) is valid for all t ∈ [t0 ,∞). For the second case,we have that E[V l(t)] ≤ ϕl(|u(t)|) for all t ∈ R≥0 . Combiningsuch two cases, we have that
REN AND XIONG: VECTOR-LYAPUNOV-FUNCTION-BASED ISS OF STOCHASTIC IMPULSIVE SWITCHED TIME-DELAY SYSTEMS 663
Part 2: Because of Definition 2 and (C.3), it obtains that
μ2N (t,t0 ) π2e2η (t−t0 ) E[V 0(t0)]
≤ μ−2N0 + 2 ( t−s )τ a π2e2η (t−t0 ) E[V 0(t0)]
≤ e(2η+ 2τ a
ln μ)(t−t0 ) μ−2N0 π2Lα2(‖ξ‖τ ).
It follows from the reverse ADT condition (C.3) that η +1τa
ln μ < 0. The remaining is along the same line as the proofof Theorem 1. As a result, the system (1) is SISS. �
Similar to Corollary 1, the following corollary is a directconsequence from the proof of Theorem 3.
Corollary 2: Consider the system (1). Assume that thereexist locally Lipschitz Lyapunov functions Ul : R≥0 ×Rnx ×M → R≥0 , l ∈ L, σ ∈ M, α1 ∈ VK∞, α2 ∈ CK∞,and ϕl ∈ K∞, positive matrices λ := diag(λ1 , . . . , λL ) andμ := diag(μ1 , . . . , μL ), and nonnegative matrices P :=[pij ]L×L ,Q := [qij ]L×L , and R := [rij ]L×L such that (μ +R)E ≺ E, (λ + P + Q)E � E, the conditions (A.1) and (C.3)hold and
D.1) for all l ∈ L, σ ∈ M and all t ∈ R≥t0 \T , U l(t) ≥ϕl(|u(t)|) implies
2LUl(t, xt , σ) ≤ λlU l(t) +L∑
j=1
pljU j (t)
+L∑
j=1
qlj sup−τ≤s≤0
U j (t + s)
D.2) for all l ∈ L and all k ∈ N>0 , U l(t) ≥ ϕl(|u(t)|) im-plies
U l(tk ) ≤ μlU l(t−k ) +L∑
j=1
rljU j (t−k )
then the system (1) is SISS.In Theorem 3, the time-delay items play a negative role in
stability analysis, which is shown from the derived reverse ADTcondition (C.3). If the time-delay items have the positive effectson the stability analysis, then the following theorem presentssufficient conditions for the SISS of the system (1) in this case.
Theorem 4: Consider the system (1). Assume that thereexist locally Lipschitz Lyapunov functions Vl : R≥0 ×Rnx ×M → R≥0 , l ∈ L, σ ∈ M, α1 ∈ VK∞, α2 ∈ CK∞,and ϕl ∈ K∞, positive matrices λ := diag(λ1 , . . . , λL ) andμ := diag(μ1 , . . . , μL ), and nonnegative matrices P :=[pij ]L×L ,Q := [qij ]L×L , and R := [rij ]L×L such that (μ +R)E ≺ E, λ + P − Q is a nonsingular M -matrix, the condi-tions (A.1) and (C.2) hold and
E.1) for all l ∈ L, σ ∈ M and all t ∈ R≥t0 \T , V l(t) ≥ϕl(|u(t)|) implies
LVl(t, xt , σ) ≤ λlV l(t) +L∑
j=1
plj
√
V l(t)V j (t)
−L∑
j=1
qlj sup−τ≤s≤0
√
V l(t)V j (t + s)
E.2) for all l ∈ L and all k ∈ N>0 , V l(t) ≥ ϕl(|u(t)|) im-plies
V l(tk ) ≤ μlV l(t−k ) +L∑
j=1
rlj
√
V l(t−k )V j (t−k )
E.3) τa < minl∈Lln(μl +
∑ Lj = 1 rl j )
−η , where η> η, η := sup{ω> 0 : −2πlω + πlλl +
∑Lj=1 plj πj −
∑Lj=1 qlj πj e
ω τ <0,
l ∈ L} with π ∈ ΩM (μ + P − Q) and maxl∈L{πl}≥ 1
then the system (1) is SISS.Proof: Since λ + P − Q is a nonsingular M -matrix, there
exists a π ∈ ΩM (λ + P − Q) with maxl∈L{ζl} ≥ 1 such thatπlλl +
∑Lj=1 pljπj −
∑Lj=1 qlj πj > 0 for all l ∈ L. Define
Γl(ω) := −πlω + πlλl +∑L
j=1 pljπj −∑L
j=1 qlj πj eωτ . Ob-
viously, Γl(0) > 0 and Γl(ω) → −∞ as ω → ∞. In addition,Γ′
l(ω) := −πl − τ∑L
j=1 qlj πj eωτ < 0 for all ω > 0. As a re-
sult, there exists a unique ωl > 0 such that Γl(ωl) = 0. Thus,we have that η = maxl∈L{ωl}.
Similar to the proof of Theorem 3, it obtains from (E.1) and(E.2) that for all l ∈ L and all t ∈ R≥t0 \T ,
E[LVl(t, xt , σ)] ≤ λl E[V l(t)]
+L∑
j=1
pij
√
E[V l(t)]E[V j (t)]
−L∑
j=1
qij sup−τ≤s≤0
√
E[V l(t)]E[V j (t + s)] (41)
and that for all l ∈ L and k ∈ N>0 , (34) holds with μ + R − I <0. In the following, we prove that for all l ∈ L and η > η,
j=1 rlj}.Following the similar fashion of the proof of Theorem 1,
define U l(t) :=√
2V l(t). Thus, (42) equals to
E[U l(t)] ≤ maxl∈L
{πl}√
2μN (t,t0 )eη (t−t0 ) E[V 0(t0)]1/2
=: maxl∈L
{Al(t − t0)}. (43)
To prove (42), we consider following two cases: (42) doesnot hold in certain impulsive switching interval and (42) doesnot hold at certain impulsive switching time. For the first case,if E[U l(t)] ≤ Al(t − t0) does not hold in (tk , tk+1), then thereexist an � ∈ L and a t∗ ∈ (tk , tk+1) such that
E[U �(t∗)] = A�(t∗ − t0) (44)
E[U �(t)] > A�(t − t0), t ∈ (t∗, t∗ + Δt) (45)
where Δt > 0 is arbitrarily small. Therefore, it obtains that
D+ E[U �(t∗)] ≥ D+A�(t∗ − t0). (46)
664 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 64, NO. 2, FEBRUARY 2019
On the other hand, it follows from (E.3), (41), (43), and (44)that
D+A�(t∗ − t0) = ηπ�
√2μN (t∗,t0 )eη (t∗−t0 ) E[V 0(t0)]1/2
>
⎛
⎝λ�
2π� +
L∑
j=1
p�j
2πj −
L∑
j=1
qlj πj eωτ
⎞
⎠
×√
2μN (t∗,t0 )eη (t∗−t0 ) E[V 0(t0)]1/2
≥ λ�
2E[U �(t∗)] +
L∑
j=1
p�j
2E[U j (t∗)]
−L∑
j=1
q�j
2sup
−τ≤s≤0E[U j (t∗ + s)]
≥ D+ E[U �(t∗)]
which contradicts with (46). Therefore, it concludes that (43)[i.e., (42)] holds for all t ∈ (tk , tk+1).
The remaining is along the same line as the proof ofTheorem 3. As a result, the system (1) is SISS, and the proof iscompleted. �
Corollary 3: Consider the system (1). Assume that thereexist locally Lipschitz Lyapunov functions Ul : R≥0 ×Rnx ×M → R≥0 , l ∈ L, σ ∈ M, α1 ∈ VK∞, α2 ∈ CK∞,and ϕl ∈ K∞, positive matrices λ := diag(λ1 , . . . , λL ) andμ := diag(μ1 , . . . , μL ), and nonnegative matrices P :=[pij ]L×L ,Q := [qij ]L×L , and R := [rij ]L×L such that (μ +R)E ≺ E, λ + P − Q is a nonsingular M -matrix, the con-ditions (A.1), (D.2), and (E.3) hold, and for all l ∈ L and allt ∈ R≥t0 \T , U l(t) ≥ ϕl(|u(t)|) implies
2LUl(t, xt , σ) ≤ λlU l(t) +L∑
j=1
pljU j (t)
−L∑
j=1
qlj sup−τ≤s≤0
U j (t + s)
then the system (1) is SISS.Remark 10: Observe from Theorems 3 and 4 that the es-
sential difference lies in the effects of the time-delay items onstability analysis. In (C.1), the time-delay items play negativeroles in the decrease of the Lyapunov functions, thereby havingnegative effects on the stability analysis. The contrary scenariooccurs in (E.1), which implies the positive effects of the timedelays on the stability analysis. Because the time-delay itemshave different effects on the stability analysis, the upper boundof the ADT are different in Theorems 3 and 4. The requirementon the ADT in Theorem 3 is more strict. In general, the upperbound of the ADT in Theorem 4 is usually larger than that inTheorems 3. Note that even if there are no time-delay items in(E.1), the stability analysis could also be proceeded along thesame fashion, and it follows from (E.3) that the upper boundof the ADT becomes smaller, which however is still larger thanthat in Theorem 3. In addition, both of the upper bounds of theADT in Theorems 3 and 4 depend on the time delays. More-over, the larger the time delays, the smaller the upper bound ofthe ADT in Theorem 3, whereas the larger the upper bound ofthe ADT in Theorem 4. See Example 1 in Section V for the
comparisons between the upper bounds of the ADT obtainedfrom Theorems 3 and 4. �
A. Stability Analysis via Comparison Principle
Similar to Section III-A, we present the relationships betweenthe vector Lyapunov function approach and the approach basedon the comparison principle in the stable discrete dynamic case.
To this end, we begin with the equivalence between the sta-bility conditions in Corollary 2 (or Corollary 3) and those inTheorem 3 (or Theorem 4). Such equivalence is obvious. Sim-ilar to Proposition 1, we discuss the relationship between thestability conditions in Corollaries 2 and 3 and those in the pre-vious works using the comparison principle. Analogous to (26)and (27) and using [27, Proposition 2.6], the conditions (D.1)and (D.2) in Corollary 2 could be rewritten as the followingvector form:
2LU(t, xt , σ) � (λ + P )U(t) + Q sup−τ≤s≤0
U(t + s)
+ ϕ(|u(t)|), t ∈ R≥t0 \T (47)
U(t) � (μ + R)U(t−) + ϕ(|u(t−)|), t ∈ T (48)
where ϕ(v) := (ϕ1(v), . . . , ϕL (v)), ϕl ∈ K∞, and l ∈ L. As aresult, the corresponding comparison system Q is given by
⎧⎪⎨
⎪⎩
2D+y(t) = (λ + P )y(t) + Q sup−τ≤s≤0 y(t + s)
+ϕ(|u(t)|) + δ, t ∈ R≥t0 \Ty(t) = (μ + R)y(t−) + ϕ(|u(t−)|), t ∈ T
where y(t) ∈ RL is the system state, and δ > 0 is arbitrary.Based on Lemma 1, we obtain the following proposition. Theproof follows the similar line as the proof of Proposition 1, andhence, omitted here.
Proposition 2: Consider the system (1). Assume that thereexist a locally Lipschitz Lyapunov function U : R≥0 × Rnx ×M → RL
≥0 , α1 ∈ VK∞, α2 ∈ CK∞, and ϕl ∈ K∞, positivematrices λ := diag(λ1 , . . . , λL ) and μ := diag(μ1 , . . . , μL ),and nonnegative matrices P := [pij ]L×L , Q := [qij ]L×L , andR := [rij ]L×L such that (μ + R)E ≺ E, λ + P − Q is a non-singular M -matrix, the condition (A.1) and the inequalities (47)and (48) hold. Suppose that the comparison system Q has aunique solution. If U(t0) � y(t0), then SISS of the comparisonsystem Q implies SISS of the system (1).
Accordingly, along the similar fashion, the correspondingcomparison system could be constructed from Corollary 3.Based on the constructed comparison system, the relationshipbetween the stability properties of the comparison system andthe stability properties of the system (1) could be established.In the following, the counterpart of Theorem 2 is given, whichprovides a connection between the obtained conditions basedon the vector Lyapunov function and the stability conditions inthe previous works [5] using the scalar Lyapunov function; seealso Remark 8 mentioned earlier.
B. Stability Analysis via Scalar Lyapunov Functions
In this subsection, the relationship between the vector Lya-punov function approach and the scalar Lyapunov function ap-proach are established for the stable discrete dynamic case.Similar to Theorem 2, the following theorem presents the sta-
REN AND XIONG: VECTOR-LYAPUNOV-FUNCTION-BASED ISS OF STOCHASTIC IMPULSIVE SWITCHED TIME-DELAY SYSTEMS 665
bility conditions via the scalar Lyapunov function, which istransformed from the vector Lyapunov function.
Theorem 5: Consider the system (1). Assume that thereexist locally Lipschitz Lyapunov functions Vl : R≥0 ×Rnx ×M → R≥0 , l ∈ L, σ ∈ M, α1 ∈ VK∞, α2 ∈ CK∞,and ϕl ∈ K∞, positive matrices λ := diag(λ1 , . . . , λL ) andμ := diag(μ1 , . . . , μL ), and nonnegative matrices P :=[pij ]L×L ,Q := [qij ]L×L ,R := [rij ]L×L such that (μ + R)E ≺E, (λ + P + Q)E � E, the conditions (A.1), (C.1), and (C.2)hold, and
τa <− ln ξ3
ξ1 + ξ2ξ−N03
(49)
where
ξ1 := maxl∈L
⎧⎨
⎩λl +
12
L∑
j=1
(plj + qlj + pjl)
⎫⎬
⎭
ξ2 :=12
maxl∈L
⎧⎨
⎩
L∑
j=1
qjl
⎫⎬
⎭,
ξ3 := maxl∈L
⎧⎨
⎩μl +
12
L∑
j=1
(rlj + rjl)
⎫⎬
⎭
then the system (1) is SISS.Proof: Similar to the proof of Theorem 2, it follows from
[27, Proposition 2.6] and the AM-GM inequality that for alll ∈ L, σ ∈ M and all t ∈ R≥t0 \T , there exist ϕl ∈ K∞, l ∈ L,
E[LVl(t, xt , σ)] ≤ λl E[V l(t)]+L∑
j=1
plj
√
E[V l(t)]E[V j (t)]
+L∑
j=1
qlj sup−τ≤s≤0
√
E[V l(t)]E[V j (t + s)] + ϕl(|u(t)|)
≤
⎛
⎝λl +12
L∑
j=1
plj +12
L∑
j=1
qlj
⎞
⎠E[V l(t)]+12
L∑
j=1
plj E[V j (t)]
+12
L∑
j=1
qlj sup−τ≤s≤0
E[V j (t + s)] + ϕl(|u(t)|) (50)
and for all l ∈ L and all k ∈ N>0 ,
E[V l(tk )] ≤ μl E[V l(t−k )] +L∑
j=1
rlj
√
E[V l(t−k )]E[V j (t−k )]
+ ϕl(|u(t−k )|)
≤
⎛
⎝μl +12
L∑
j=1
rlj
⎞
⎠E[V l(t−k )] +12
L∑
j=1
rlj E[V j (t−k )]
+ ϕl(|u(t−k )|). (51)
Therefore, we transform the coupled items into the decoupleditems.
Define V(t, x, σ) :=∑L
l=1 Vl(t, x, σ), V(t) :=∑L
l=1 V l(t)and ϕ(v) :=
∑Lj=1 ϕl(v). Based on (50)–(51), it follows that
for all l ∈ L, σ ∈ M and all t ∈ R≥t0 \T ,
E[LV(t, xt , σ)]≤maxl∈L
⎧⎨
⎩λl +
12
L∑
j=1
(plj +qlj +pjl)
⎫⎬
⎭E[V(t)]
+12
maxl∈L
⎧⎨
⎩
L∑
j=1
qjl
⎫⎬
⎭sup
−τ≤s≤0E[V(t + s)] + ϕ(|u(t)|)
(52)
and for all l ∈ L and all k ∈ N>0 ,
E[V(tk )] ≤ maxl∈L
⎧⎨
⎩μl +
12
L∑
j=1
(rlj + rjl)
⎫⎬
⎭E[V(t−k )]
+ ϕ(|u(t−k )|). (53)
Based on the reverse ADT condition (49) and (52)–(53), itconcludes from [5, Th. 2] that the system (1) is SISS. �
Remark 11: According to the decoupling and summing tech-niques and [5, Th. 2], the SiISS and weighted ISS could alsobe obtained; see also Remark 8. Furthermore, if P,Q, and Rare symmetric and π = E, then we observe that with the smalltime delay, the upper bound of the ADT in Theorem 3 is largerthan that in Theorem 5; see Example 1 in Section V. However,if the time delay increases, then the upper bound of the ADT inTheorem 3 becomes gradually smaller than that in Theorem 5.The main reason lies in that the upper bound of the ADT inTheorem 3 depends on the time delay (see also Remark 10),whereas the upper bound of the ADT in Theorem 5 does not.�
V. ILLUSTRATIVE EXAMPLES
In this section, two numerical examples are presented to il-lustrate the developed results in the previous sections.
Example 1: Consider a two-neuron stochastically perturbedneural network S given in [20] and [21] with impulsive switch-ing effects
dx(t) = [Aσ (t)x(t) + Bσ (t)f(x(t)) + Cσ (t)f(xt)
+ Fσ (t)u(t)]dt + Dσ (t)x(t)dB(t), t ∈ R≥0\T ,
x(t) = Hσ (t−)x(t−), t ∈ T
where x(t) ∈ R2 , σ(t) ∈ {1, 2}, τ(t) ∈ [0, 1], and T isa given impulsive switching time sequence. Assume thatAσ (t) ,Dσ (t) , and Hσ (t) are diagonal matrices and that there ex-ist θi > 0 such that 0 ≤ (fi(z1) − fi(z2))/(z1 − z2) ≤ θi forall z1 , z2 ∈ R and i = 1, 2.
Define Vi(t, x(t), σ(t)) := x2i (t) for i, σ ∈ {1, 2} and
f(x(t)) = (f1(x1(t)), f2(x2(t))) = (tanh(x1(t)), tanh(x2(t))). In this example, xt := (x1(t − τ1), x2(t − τ2)), that is,the different state elements are allowed to have different timedelays. In the following, let us consider three different cases.
666 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 64, NO. 2, FEBRUARY 2019
Fig. 1. Given τ1 = 0.8, τ2 = 1, x0 = (5,−3), and τa = 0.95, stateresponse of the system S in case I.
Case I: Set H1 = H2 = 1.3I , D1 = I , D2 = diag(√
2, 2)and
A1 =
[−4 0
0 −4.5
]
, A2 =
[−6 0
0 −7
]
, B1 =
[0.2 2/3
0.6 1
]
B2 =
[0.6 0.8
0.8 0
]
, C1 =
[0.4 0.5
0.6 1/3
]
, C2 =
[0.5 0.9
0.9 0.7
]
.
Obviously, the continuous dynamics is stable and the discretedynamics is unstable. Therefore, it obtains from computationthat LV1(t, xt , 1) ≤ −5.6x2
(t) + 1.6x2(t)x1(t) + 1.8x2(t)x1 (t − τ1) + 1.4x2(t)x2(t −τ2) + u2(t). Based on Theorem 1, we have that η = 0.6827,which implies that the ADT τ 1
a > 0.7686.1 Let τa = 0.95 andu(t) = sin t. Given the initial state x0 = (5,−3) and τ1 = 0.8,τ2 = 1, the state response S is presented in Fig. 1.
On the other hand, if the Lyapunov function isscalar and chosen as V(t, x(t), σ(t)) = V1(t, x(t), σ(t)) +V2(t, x(t), σ(t)) = x�(t)x(t), then it obtains from Theorem 2that � = 1.1856 and that the ADT satisfies τ 2
a > 0.4426. Thatis, η < �; the lower bound of τ 2
a is smaller than that of τ 1a ,
which means that the feasible region for τ 2a is larger. In addi-
tion, it follows from Theorems 1 and 2 that the larger the timedelays, the larger the lower bounds of τ 1
a and τ 2a .
Case II: Set H1 = H2 = diag(0.6, 0.8), D1 = diag(1,√
2),D2 = diag(
√3, 1), and
A1 =
[3 0
0 2
]
, A2 =
[2 0
0 1
]
, B1 =
[1 0.5
0.2 0
]
B2 =
[0.5 0.2
0.7 0.4
]
, C1 =
[−2 −1
−0.8 −1.5
]
,
C2 =
[−1 −1.5
−1 −0.5
]
.
That is, the continuous dynamics is unstable while the discretedynamics is stable. As a result, it obtains from computation that
1In Example 1, the notation τ 1a is the same as τa . The superscript in τa is to
emphasize which theorem the ADT comes from.
Fig. 2. Given τ1 = 0.8, τ2 = 1, x0 = (5,−5), and τa = 0.15, stateresponse of the system S in case II.
2(t) + 1.4x1(t)x2(t) − 2x2(t)x1(t − τ1) − x2(t)x2(t − τ2) + u2(t). Thatis, the time-delay items play the positive roles in stabilityanalysis. According to Theorem 4, it obtains that η = 0.6141,which implies that the ADT τ 4
a < 0.7268. Choose τa = 0.15and given the initial state x0 = (5,−5) and τ1 = 0.8, τ2 = 1,the state response of the system S is presented in Fig. 2.
Case III: Set H1 = H2 = 0.7071I , D1 = D2 = I and
A1 =
[0.5 0
0 0.3
]
, A2 =
[0.4 0
0 2/7
]
, B1 =
[0.2 2/3
0.4 1/3
]
B2 =
[0.1 0.2
3/7 0
]
, C1 =
[0.2 1/3
0.1 0.3
]
, C2 =
[0.5 0.2
0.3 0.2
]
.
In this case, the continuous dynamics is unstable while thediscrete dynamics is stable. After the detailed computation, wehave that LV1(t, xt , 1) ≤ 3.4x2
τ2) + u2(t). That is, the time-delay items play the negativerole in stability analysis. According to Theorem 3, we havethat η = 3.4100, which indicates that the ADT τ 3
a < 0.2033.Choose τa = 0.1 and given the initial state x0 = (5,−3) andτ1 = τ2 = 0.2, the state response of the system S is shownin Fig. 3.
Furthermore, if the Lyapunov function is scalar and chosenas V(t, x(t), σ(t)) = V1(t, x(t), σ(t)) + V2(t, x(t), σ(t)) =x�(t)x(t), then we have from Theorem 5 that the ADT satisfiesτ 5a < 0.1014, which implies that the upper bounds of τ 3
a islarger than that of τ 5
a . On the other hand, the upper bound ofthe ADT obtained from Theorem 4 is much larger than thoseobtained from Theorems 3 and 5. As a result, the time-delayitems have great effects on the upper bound of the ADT. Inaddition, the larger the time delays, the smaller the upper boundof τ 3
a , but the larger the upper bound of τ 4a . On the contrary, the
time delay has no effects on τ 5a .
Example 2: Consider a switched chaos-based secure com-munication system [19] consisting of two Lorenz chaotic
REN AND XIONG: VECTOR-LYAPUNOV-FUNCTION-BASED ISS OF STOCHASTIC IMPULSIVE SWITCHED TIME-DELAY SYSTEMS 667
Fig. 3. Given τ1 = τ2 = 0.2, x0 = (5,−3), and τa = 0.1, state re-sponse of the system S in case III.
z(t) = z(t−) + Dσ (t)(x(t−) − z(t−)), t = rk , k ∈ N
where Aσ (t) and Bσ (t) are the matrices of appropriatedimensions; Θσ (t) and Θσ (t) are continuous functions intheir domains of definition, respectively. From [19], chooseΘσ (t)(z(t), x(t − τ), z(t − τ)) := Θσ (t)(z(t)) + Fσ (t)Kσ (t)(x(t − τ) − z(t − τ)).
Define by e(t) := x(t) − z(t) the synchronization error be-tween the transmitter system and the receiver system. Then, theerror dynamics, denoted by E , is given by
Consider the following two subsystems with the parametersas follows: u(t) = sin(t), B1 = B2 = C1 = C2 = F2K2 = I ,F1K1 = 5I , D1 = D2 = (−1 +
√0.5)I ,
A1 =
⎡
⎢⎣
−10 10 0
28 −1 0
0 0 −8/3
⎤
⎥⎦ , Θ1(x) =
⎡
⎢⎣
0
−x1x3
x1x2
⎤
⎥⎦
A2 =
⎡
⎢⎣
−27/14 9 0
1 −1 1
0 −14.286 0
⎤
⎥⎦ ,
Θ2(x) =
⎡
⎢⎣
−27 sat(x1)/28
0
0
⎤
⎥⎦
where sat(x1) is a saturation function and defined as sat(x1) =12 (|x1 + 1| − |x1 − 1|). Given the initial states (−5, 10,−10)
Fig. 4. Under the initial state (−5, 10,−10), state response of the trans-mitter system of subsystem 1.
Fig. 5. Under the initial state (−3, 1, 4), state response of the transmit-ter system of subsystem 2.
and (−3, 1, 4), the state responses of the transmitter systemsof two subsystems are shown in Figs. 4 and 5, respectively.Obviously, both of them are Lorenz attractors. Observe fromFig. 4 that
Θ1(x(t)) ≤
⎡
⎢⎣
0 0 0
60 0 30
40 30 0
⎤
⎥⎦ x(t).
Moreover, Θ2(x(t)) − Θ2(z(t)) ≤ 27|e1(t)|/28.Choose the vector Lyapunov function as V (t, e(t), σ(t))
= (V1(t, e(t), σ(t)), V2(t, e(t), σ(t))) :=((e1(t) + e2(t))2 , (e2(t) + e3(t))2). According to the aforementioned subsys-tems and assumptions, it follows from computation thatLV1 (t, et , 1) ≤58(e1(t)+e2(t))2 + 60(e1(t) +e2(t))(e2(t) +e3(t)) − 10(e1(t) + e2(t))(e1(t − τ) + e2(t − τ)) + 4u2(t);LV2(t, et , 1) ≤ 88
3 (e2(t) + e3(t))2 + 128(e1(t) + e2(t))(e2(t)+ e3(t))−10(e2(t) + e3(t))(e2(t − τ) +e3(t − τ)) + 4u2(t),LV1(t, et , 2) ≤ 15(e1(t) + e2(t))2 + 2(e1(t) + e2(t)) (e2(t)+ e3(t)) − 2(e1(t) + e2(t))(e1(t − τ) + e2(t − τ)) + 4u2(t),and LV2(t, et , 2) ≤3(e2(t) + e3(t))2 + 2(e1(t) +e2(t))(e2(t)+ e3(t)) − 2(e2(t) + e3(t))(e2(t − τ) + e3(t − τ)) + 4u2(t).Based on Theorem 4, we have that η = 3.3897, and hence,τ 4a < 0.2045. Pick τa = 0.2 and given the initial state
e0 = (−2, 4,−5) and τ = 0.2, the state response of the errordynamic system E is shown in Fig. 6.
On the other hand, if the Lyapunov function is scalar andclassical quadratic, that is, V(t, e(t), σ(t)) := e�(t)e(t). Wehave that for all t ∈ R≥t0 \T , LV(t, et , 1) ≤ 128e�(t)e(t) +e�t et + 3u�(t)u(t) and LV(t, et , 2) ≤ 28e�(t)e(t) + e�t et +3u�(t)u(t). As a result, the result in Theorem 4 cannot be used.It follows from Theorem 5 that τ 5
a < 0.0053. Comparing theADT conditions obtained via the vector Lyapunov function and
668 IEEE TRANSACTIONS ON AUTOMATIC CONTROL, VOL. 64, NO. 2, FEBRUARY 2019
Fig. 6. Under the initial state (−2, 4,−5), state responses of the errordynamic system E .
the scalar Lyapunov function, we find that the upper bound of τ 5a
is much smaller than that of τ 4a . Therefore, the vector-Lyapunov-
function-based stability conditions are less conservative than thescalar-Lyapunov-function-based stability conditions.
VI. CONCLUSION
In this paper, the ISS was established for stochastic impulsiveswitched time-delay systems. Based on the vector Lyapunovfunctions and the average dwell-time condition, sufficient con-ditions were derived. Both the case that the continuous dynamicsis stable and the case that the discrete dynamics is stable werestudied. The relationship between this study and the previousworks was discussed. Two numerical examples were given to il-lustrate the developed theory. Future research could be directedto the switching law design to stabilize switched systems withunstable subsystems, controller design for switched systemswith asynchronous phenomena.
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Wei Ren received the B.Sc. degree in informa-tion and computation science from Hubei Uni-versity, Wuhan, China, in 2011. He is currentlyworking toward the Ph.D. degree with the Univer-sity of Science and Technology of China, Hefei,China.
From April 2017 to April 2018, he was ajoint Ph.D. student with the University of Mel-bourne, Australia. His research interests includethe fields of networked control systems, non-linear systems, stochastic systems, and hybridsystems.
Junlin Xiong (M’11) received the B.Eng. degreein mineral process engineering and and M.Sc.degree in operation research and control theoryfrom Northeastern University, Shenyang, China,in 2000 and 2003, respectively, and the Ph.D.degree in mechanical engineering from the Uni-versity of Hong Kong, Hong Kong, in 2007.
From November 2007 to February 2010, hewas a Research Associate with the Universityof New South Wales, Australian Defence ForceAcademy, Canberra, Australia. In March 2010,
he joined the University of Science and Technology of China, Hefei,China, where he is currently a Professor with the Department of Au-tomation. His current research interests include the fields of Marko-vian jump systems, networked control systems, and negative imaginarysystems.
