Stability and L2 -gain analysis for impulsive delay systems: An impulse-time-dependent discretized Lyapunov functional method

Automatica 86 (2017) 129–137

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Brief paper

Stability and L2 -gain analysis for impulsive delay systems: An impulse-time-dependent discretized Lyapunov functional method✩ Wu-Hua Chen a , Zhen Ruan a , Wei Xing Zheng b a b

College of Mathematics and Information Science, Guangxi University, Nanning, Guangxi, 530004, PR China School of Computing, Engineering and Mathematics, Western Sydney University, Sydney, NSW 2751, Australia

article

info

Article history: Received 1 December 2016 Received in revised form 9 June 2017 Accepted 4 August 2017

Keywords: Impulsive systems Time delay Exponential stability L2 -gain Discretized Lyapunov functional

a b s t r a c t The problems of exponential stability and L2 -gain for a class of time-delay systems with impulsive effects are studied. The main tool used is the construction of an impulse-time-dependent complete Lyapunov functional. By dividing the impulse interval and delay interval into several segments, the matrix functions of this functional are chosen to be continuous piecewise linear. Moreover, an impulse-time-dependent weighting factor is introduced to coordinate the dynamical behavior of the nondelayed and integral terms of this functional along the trajectories of the system. By applying this functional, delay-dependent sufficient conditions for exponential stability and L2 -gain are derived in terms of linear matrix inequalities. As by-products, new delay-independent sufficient conditions for the same problems are also derived. The efficiency of the proposed results is illustrated by numerical examples. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction As a subclass of hybrid systems, impulsive systems have gained significant research interest due to their advantages in characterizing instantaneous jump phenomena occurring in many real evolutionary processes. Such systems have numerous applications in mechanics, radio engineering, biology, biotechnology, control of industrial robots and so on (Bainov and Simeonov, 1989; Yang, 2001). Another source of motivation for studying impulsive systems comes from the rapidly developing field of networked control systems (Jentzen, Leber, Schneisgen, Berger, & Siegmund, 2010; Tolić & Hirche, 2017) and sampled-data control (Heemels, Donkers, & Teel, 2013; Liu & Fridman, 2012; Naghshtabrizi, Hespanha, & Teel, 2008, 2010). The closed-loop performance of networked/sampled-data control systems can be analyzed in the impulsive system paradigm. In recent years, aiming at exploiting the hybrid structure characteristics of impulsive systems (Goebel, Sanfelice, & Teel, 2012), several efficient analysis tools have been developed: e.g., Lyapunov ✩ This work was supported in part by the National Natural Science Foundation of China under Grants 61633011 and 61573111, the Guangxi Natural Science Foundation under Grants 2015GXNSFAA139003, the Innovation Project of Guangxi Graduate Education ( YCSW2017049), and the Australian Research Council under Grant DP120104986. The material in this paper was not presented at any conference. This paper was recommended for publication in revised form by Associate Editor Emilia Fridman under the direction of Editor Ian R. Petersen. E-mail addresses: [email protected] (W.-H. Chen), [email protected] (Z. Ruan), [email protected] (W.X. Zheng). http://dx.doi.org/10.1016/j.automatica.2017.08.023 0005-1098/© 2017 Elsevier Ltd. All rights reserved.

functions (Hespanha, Liberzon, & Teel, 2008), Lyapunov functionals (Naghshtabrizi et al., 2008), impulse-time-dependent Lyapunov functions (Chen, Li, & Lu, 2011), looped-functionals (Briat & Seuret, 2012a, 2012b), clock-dependent Lyapunov functions (Briat, 2013), polytopic embedding method (Hetel, Daafouz, Tarbouriech, & Prieur, 2013), and set-introduced polyhedral Lyapunov functions (Fiacchini & Morˇarescu, 2016). On the other hand, timedelays commonly appear in various practical systems, which are often a source of instability and poor performance. The interaction of impulses and time-delays complicates stability analysis. During the last decade, many efforts have been devoted to extending the stability theory developed for continuous-time time-delay systems to their counter parts for impulsive time-delay systems. Some important and interesting results have been proposed, for example, the Bohl–Perron type theorem for linear impulsive delay differential equations (Anokhin, Berezansky, & Braverman, 1995), the impulse-type Razumikhin theorems (Chen & Zheng, 2011; Liu & Ballinger, 2001), the impulsive delay differential inequality based techniques (Li & Song, 2017; Yang & Xu, 2007), and the impulsetime-dependent Lyapunov functional based method (Chen, Li, & Lu, 2013). Despite all this progress, one may find that the existing methods for stability analysis consider the delayed terms to be always detrimental to the stability of the system. However, for some impulsive delay systems, the delayed terms with small timedelay could contribute to stability. In this case, naturally there arises a question: how to establish a stability criterion such that the stabilizing role of the delayed term with small time-delay can be captured. For continuous time-delay systems, such stability

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analysis is referred to as delay-dependent stability analysis. A commonly used technique is to apply the Newton–Leibnitz formula for representing the relation among the state, the delayed state, and the state derivative (Fridman, 2014). Note that in the presence of impulses, the corresponding Newton–Leibnitz formula contains several jump terms, which are difficult to be handled using Lyapunov functional (Chen & Zheng, 2009a). In this paper, inspired by the impulse-time-dependent Lyapunov functional introduced in Chen et al. (2013) and the discretized Lyapunov functional method developed in Fridman and Shaked (2000) and Gu (1997), we propose a novel construction of Lyapunov functional for a class of impulsive delay systems. In the proposed method, the time-dependent complete Lyapunov functional is divided into two parts: the nondelayed term and the integral terms. In the part of the nondelayed term, by dividing the impulse interval into several segments, the positive definite matrix function is chosen to be a time-dependent convex combination of two constant positive definite matrices within each segment. In the part of the integral terms, following the discretization procedure of Gu (1997), the matrix functions are selected to be continuous piecewise linear. Moreover, as in Chen et al. (2013), an impulsetime-dependent weighting factor is introduced to coordinate the nondelayed and integral terms. The introduced impulse-timedependent discretized Lyapunov functional synthesizes the merits of the impulse-time-dependent Lyapunov functional and the discretized Lyapunov functional. First, coordinated by the impulsetime-dependent weighting factor, the new Lyapunov functional can be made to decrease along the system trajectories by choosing appropriate parameter matrices. Second, the delay-dependent stability condition via the complete Lyapunov functional and discretization reduces the conservatism that stems from the use of impulse-type Newton–Leibnitz formula. The finite L2 -gain problem is also solved applying the new type Lyapunov functional. As by-products, we obtain new delay-independent results for exponential stability and L2 -gain. The remainder of this paper is arranged as follows. Section 2 describes the model of the impulsive delay system and introduces the relevant notation. The problems of exponential stability and L2 -gain are analyzed via the introduced impulse-time-dependent discretized Lyapunov functional in Section 3 and Section 4, respectively. Section 5 provides three numerical examples to demonstrate the applicability of the obtained results. Finally, conclusions are drawn in Section 6. 2. Problem formulation

where x(tk+ ) = x(tk ) = limh→0+ x(tk + h), x(tk− ) = limh→0− x(tk + h), k = 1, 2, . . . , and 0 < t1 < t2 < · · · < tk < · · · (tk → ∞ as k → ∞). This paper aims at developing an impulse-time-dependent discretized Lyapunov functional method for exploring the effects of impulse and delay to exponential stability and L2 -gain of system (1). Now the definitions of exponential stability and L2 -gain are given. Definition 1. For a given class S of impulse time sequences, the zero solution of (1) with w (t) = 0 is said to be uniformly globally exponentially stable (UGES) over S if there exist two positive scalars M and ν such that, for any impulse time sequence {tk } ∈ S and any initial function φ ∈ PC ([−τ , 0], Rn ), the solution x(t) = x(t , t0 , φ ) of (1) with w (t) = 0 satisfies ∥x(t)∥ ≤ M ∥φ∥τ exp(−ν (t − t0 )), ∀ t ≥ t0 . Furthermore, ν is called the exponential convergence rate. Definition 2. For a given class S of impulse time sequences and a positive scalar γ , system (1) is said to have L2 -gain γ over S, if under the zero initial condition φ (θ ) = 0, θ ∈ [−τ , 0], for any impulse time sequence {tk } ∈ S, it holds that t

∫

t

∫

z T (s)z(s)ds ≤ γ 2

wT (s)w (s)ds, ∀ t ≥ 0. 0

0

In this paper, for given positive scalars σ0 and σ1 with σ0 ≤ σ1 , the notation S(σ0 , σ1 ) is used to denote the class of impulse time sequences {tk } satisfying σ0 ≤ tk − tk−1 ≤ σ1 for all k ∈ N. According to Briat and Seuret (2012a, 2012b), S(σ0 , σ1 ) represents a ranged dwell-time condition. 3. Stability analysis In this section we will focus on the internal stability analysis of system (1). For this purpose, we consider system (1) with w (t) = 0, that is, x˙ (t) = A0 x(t) + A1 x(t − τ ), t ̸ = tk , ∆x(t) = C0 x(t − ), t = tk , x(t0 + θ ) = φ (θ ), t0 = 0, θ ∈ [−τ , 0],

(2)

where {tk } ∈ S(σ0 , σ1 ).

In the sequel, if not explicitly stated, matrices are assumed to have compatible dimensions. The notation M > 0 is used to denote a symmetric positive-definite matrix. ∥ · ∥ stands for the Euclidean norm for vectors or the spectral norm for matrices. N denotes the set of positive integers. For two integers n1 and n2 with n1 ≤ n2 , the notation n1 , n2 represents the set of {n1 , n1 + 1, . . . , n2 }. For τ > 0, let PC ([−τ , 0], Rn ) denote the set of piecewise right continuous function φ : [−τ , 0] → Rn with the norm defined by ∥φ∥τ = sup−τ ≤θ ≤0 ∥φ (θ )∥. If x ∈ PC ([−τ , b), Rn ) with b > 0, then for each t ∈ [0, b), we define xt ∈ PC ([−τ , 0], Rn ) by xt (θ ) = x(t + θ ) for −τ ≤ θ ≤ 0. Consider a class of time-delay systems with impulsive effects described by the following state equation x˙ (t) = A0 x(t) + A1 x(t − τ ) + H1 w (t), t ̸ = tk ,

∆x(t) = C0 x(t − ), t = tk , z(t) = Ex(t) + H2 w (t), x(t0 + θ ) = φ (θ ), t0 = 0, θ ∈ [−τ , 0],

output. A0 , A1 , H1 , H2 , E and C0 , are known constant matrices with appropriate dimensions. τ > 0 denotes the state delay, and φ ∈ PC ([−τ , 0], Rn ) is the initial condition of the state. ∆x(tk ) = x(tk+ ) − x(tk− ) describes the state jumping at impulse instant t = tk ,

(1)

where x(t) ∈ Rn is the state, w (t) ∈ Rp is the disturbance input which belongs to L2 (0, ∞), and z(t) ∈ Rq is the controlled

3.1. Impulse-type complete Lyapunov functionals In order to construct an impulse-time-dependent Lyapunov functional, we divide the impulse interval [tk−1 , tk ) into N0 subintervals [tk,l−1 , tk,l ) of equal length δk ≜ tk,l − tk,l−1 = (tk − tk−1 )/N0 , l ∈ 1, N0 , where tk,0 = tk−1 , tk,N0 = tk , k ∈ N. Then, we introduce two piecewise linear functions associated with the impulse time sequence {tk } ∈ S(σ0 , σ1 ) as follows:

ϱ(t) = ρ10 (t) =

1

, t ∈ [tk−1 , tk ), k ∈ N,

tk − tk−1 t − tk,l−1

tk,l − tk,l−1

, t ∈ [tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N.

Inspired by the impulse-time-dependent Lyapunov functional introduced in Chen et al. (2013) and the complete Lyapunov functional introduced in Fridman and Shaked (2000) and Gu (1997), we shall find such an impulse-type complete Lyapunov functional for system (2): V (t , xt ) = V1 (t , xt ) + V2 (t , xt ) + V3 (t , xt ),

(3)

W.-H. Chen et al. / Automatica 86 (2017) 129–137

131

where Vi (t , xt ), i ∈ 1, 3, have the following form:

Proof. Using (5), (8), and (9), we have

V1 (t , xt ) = e2ν t ϕ (t)xT (t)P(t)x(t),

V (tk ) = xT (tk− )(I + C0 )T e2ν tk P0 (I + C0 )x(tk− )

V2 (t , xt ) = 2xT (t) t

∫

V3 (t , xt ) =

∫

t

eν (s+t) Q (t , s − t)x(s)ds

t −τ t

∫

e t −τ

[

+2

x (s1 )R(s1 − t , s2 − t)ds1

t −τ

≤e

t

× x(s2 )ds2 +

e

2ν s T

x (s)S(s − t)x(s)ds.

ϕ (t) =

µi µl

] Q0 (s − tk )x(s)ds + V3 (tk )

∫

, P(t) = ρ10 (t)Pl + ρ11 (t)Pl−1 ,

µi xT (tk− )PN0 x(tk− ) + 2xT (tk− )

i=1 tk

eν (s+tk ) QN0 (s − tk )x(s)ds + V3 (tk )

×

In the above, ν > 0, and for each k ∈ N, P : [tk−1 , tk ) → Rn×n , Q : [tk−1 , tk ) × [−τ , 0] → Rn×n , R(ξ , η) = RT (η, ξ ) : [−τ , 0] × [−τ , 0] → Rn×n , and S : [−τ , 0] → Rn×n are continuous. The impulse-time-dependent functions ϕ (t), P(t), and Q (t , η) take the following form: ρ10 (t)

N0 ∏

2 ν tk

t −τ

l−1 ∏

e

ν (s+tk )

tk −τ

ν (s1 +s2 ) T

∫

tk

∫

tk −τ

= e2ν tk ϕ (tk− )xT (tk− )P(tk− )x(tk− ) + 2xT (tk− ) ∫ tk × eν (s+tk ) Q (tk− , s − tk )x(s)ds + V3 (tk− ) tk −τ

= V (tk− ).

(10)

This completes the proof. ■

i=1

Q (t , η) = ρ10 (t)Ql (η) + ρ11 (t)Ql−1 (η), η ∈ [−τ , 0],

3.2. Discretization

for t ∈ [tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N,

(4)

where ρ11 (t) = 1 − ρ10 (t), µl , l ∈ 1, N0 , are positive scalars, 0 < Pl ∈ Rn×n , l ∈ 0, N0 , and Ql (η), l ∈ 0, N0 , are n×n continuous matrix functions. Note that ϕ (t) and P(t) are continuous and piecewise differentiable in (tk−1 , tk ). Moreover, for all k ∈ N, and for η ∈ [−τ1 , 0],

ϕ (tk ) = 1, ϕ (tk− ) =

N0 ∏

µi , P(tk ) = P0 , P(tk− ) = PN0 ,

i=1

Q (tk , η) = Q0 (η), Q (tk− , η) = QN0 (η).

(5) 1−j , l

∏l

Set µ ˜ 0 = 1, µ ˜l = ˆ lj = µ ˜ l−1 µ j ∈ 0, 1, i=1 µi , and µ , k ∈ N , define ρ20 (t) = l{ ∈ 1, N0 . For t ∈ [tk,l−1 , tk,l ), l ∈ 1, N 0 { ϕ (t) − µ ˜ l−1 , µ ˜l −µ ˜ l−1 µ ˜ l−1 ,

if µl ̸ = 1

if µl = 1

, and ρ30 (t) =

ϱ(t) − 1/σ1 , 1/σ0 − 1/σ1 1/σ0 ,

ρm1 (t) = 1 − ρm0 (t), m ∈ 2, 3. Then, we obtain 1

ϕ (t) =

∑

1

ρ2j (t)µ ˆ lj , ϱ(t) =

∑ ℓ=0

j=0

ρ3ℓ (t)

1

σℓ

,

if σ0 < σ1

if σ0 = σ1

. Let

Applying the discretization method proposed in Gu (1997), we divide the delay interval [−τ , 0] into N subintervals [θp , θp−1 ] of equal length h ≜ θp−1 − θp = τ /N, p ∈ 1, N. This division also divides the square [−τ , 0] × [−τ , 0] into N × N small squares [θp , θp−1 ] × [θq , θq−1 ], p, q ∈ 1, N. Each small square is further divided into two triangles. The continuous matrix functions Ql (η), l ∈ 0, N0 , and S(η) are chosen to be linear with respect to η within each subinterval and the continuous matrix function R(ξ , η) is chosen to be linear within each triangle. Then, they can be expressed in the following forms: (q)

Ql (β ) ≜ Ql (θq + hβ ) = (1 − β )Qq,l + β Qq−1,l ,

(11)

(β ) ≜ S(θq + hβ ) = (1 − β )Sq + β Sq−1 ,

(12)

S

(q)

R

(p,q)

(α, β ) ≜ R(θp + α h, θq + β h) (1 − α )Rpq + β Rp−1,q−1 + (α − β )Rp−1,q , α ≥ β , (1 − β )Rpq + α Rp−1,q−1 + (β − α )Rp,q−1 , α < β

{ =

α, β ∈ [0, 1], p, q ∈ 1, N , l ∈ 0, N0 , (6)

for t ∈ [tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N.

where Sp , Rpq = RTqp , p, q ∈ 0, N, are n × n constant matrices. Applying the above representations to Ql (η), S(η) and R(ξ , η), we can rewrite V (t) in (3) as V (t) = e

Let

(13)

2ν t

[ N ∑ ϕ (t)xT (t)P(t)x(t) + 2xT (t) VQq (t , xt ) q=1

ρ˜ ij (t) = ρ1i (t)ρ2j (t), ρ¯ ijℓ (t) = ρ1i (t)ρ2j (t)ρ3ℓ (t).

(7)

For notational brevity, set Vi (t) = Vi (t , xt ), i ∈ 1, 3, and V (t) = V (t , xt ). From the construction of V (t), along the solution of system (2), V (t) is continuous inside impulse intervals (tk−1 , tk ), k ∈ N, but may be discontinuous at impulse instants. The role of impulsetime-dependent functions ϕ (t) and ρ10 (t) is to render V (t) decrease at impulse instants tk under appropriate conditions.

+

(I + C0 )T P0 (I + C0 ) ≤

Q (q) (t , β )x(q) (t , β )dβ, 0 1

∫

(8)

(I + C0 )T Q0 (η) = QN0 (η), ∀ η ∈ [−τ , 0], k ∈ N.

(9)

Then the Lyapunov functional V (t) is non-increasing at impulse instants tk , k ∈ N.

1

∫

(

VRpq =

)T

x(p) (t , α ) R(p,q) (α, β )x(q) (t , β )dα dβ,

0 1

x(q) (t , β )

(

VSq =

µi PN0 ,

(14)

q=1

1

∫ VQq =

0

i=1

] VSq (xt ) ,

where

∫

N0

VRpq (xt ) +

N ∑

p,q=1

Proposition 1. Assume that

∏

N ∑

0

)T 1 h

S (q) (β )x(q) (t , β )dβ,

where x(q) (t , β ) = eν (θq +hβ ) hx(t + θq + hβ ), and Q (q) (t , β ) = ∑ (q) 1 i=0 ρ1i (t)Ql−i (β ), for t ∈ [tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N. In the sequel, the following notation will be used:

xˆ (q) (t , β ) =

∫ β

1

x(q) (t , s)ds,

132

W.-H. Chen et al. / Automatica 86 (2017) 129–137 β

∫

xˆ (q) (t , β ) =

− e−ντ Q (t , −τ )x(t − τ ) [ ∫ t

x(q) (t , s)ds, q ∈ 1, N ,

0

Xˆ (t , β ) = col xˆ (1) (t , β ), xˆ (2) (t , β ) + xˆ (1) (t , β ), . . . ,

(

V˙ 3 (t) = 2e2ν t xT (t)

xˆ (N) (t , β ) + xˆ (N −1) (t , β ), xˆ (N) (t , β ) ,

)

The following proposition provides an equivalent representation of V (t) in (14). Proposition 2. The Lyapunov functional V (t) in (14), with ϕ (t), P(t), (q) Ql , S (q) , and R(p,q) , defined in (4), and (11)–(13), satisfies 1 0

+e

2ν t

[ ζˆ T (t , β ) ϕ (t)P(t) ∗ N ∑

]

Qˆ (t) ˆ ζ (t , β )dβ R

VSq (xt ), t ≥ 0,

(15)

where ζˆ (t , β ) = col x(t), Xˆ (t , β ) , Qˆ (t) = [Q0 (t) Q1 (t) . . . QN (t)]

(

with Qq (t) = and

⎡

R01 R11

R00 ⎢ R10

R= ⎣

)

ρ

for t ∈ [tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N,

i=0 1i (t)Qq,l−i ,

···

···

RN0

⎤

R0N R1N ⎥

··· ··· ··· ···

RN1

···⎦

Proposition 3. For the Lyapunov functional V (t) in (14), with ϕ (t), (q) P(t), Ql , S (q) , and R(p,q) , defined in (4), and (11)–(13), its derivative ˙V (t) along the trajectory of system (2) satisfies

. V˙ (t) =

RNN

[ 1 ∑ µ ˆ P V (t) ≥ e2ν t ζˆ T (t , β ) lj l−i ρ˜ ij (t) ∗ 0 ∫

i,j=0

ˆ l−i Q ˆ R

V (t) ≥ e2ν t

1

∫

ζˆ T (t , β ) ⎣

with

ϕ (t)P(t) ∗

⎤

Qˆ (t) 1 ⎦ ζˆ (t , β )dβ R+ S h

V˙ 1 (t) = e2ν t ϕ (t) xT (t) (2ν + N0 ϱ(t) ln µl ) P(t) + N0 ϱ(t)

(

× (Pl − Pl−1 ) x(t) + 2x (t)P(t)Aζ (t) , T

)

]

(17)

eν (s−t) Q (t , s − t)x(s)ds

t −τ T

+ x (t)

∫

t

e

ν (s−t)

[

ν Q (t , s − t) + Q˜ (t , s − t)

t −τ ( × x(s)ds + xT (t) Q (t , 0)x(t)

]

In the above,

] µ ˆ lj Pl−i A1 − e−ντ QN ,l−i , −2ντ −e SN ] [ 1 Dliℓ1ı − 2(2 − ı)Q2,l−i , ı ∈ 1, 2, Dliℓı = (−1)ı−1 hAT1 Qı,l−i − e−ντ hRNı Sd = diag (S0 − S1 , S1 − S2 , . . . , SN −1 − SN ) , ⎡ ⎤ Rd11 Rd12 · · · Rd1N ⎢ Rd21 Rd22 · · · Rd2N ⎥ Rd = ⎣ , ··· ··· ··· ··· ⎦ RdN1 RdN2 · · · RdNN Ξlijℓ =

[ 1 Ξlijℓ ∗

in which Rdpq = Rp−1,q−1 − Rpq , p, q ∈ 1, N, and

Ξlij1 ℓ = µ ˆ lj

where A = [A0 A1 ], and ζ (t) = col(x(t), x(t − τ )). For t > 0, and t ̸ = tk,l , l ∈ 1, N0 , k ∈ N, the derivatives of Vi (t), i ∈ 2, 3, along the solution of system (1), are given by t

0

t ∈ (tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N.

For t ∈ (tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N, differentiating V1 (t) along the solution of system (1) yields

∫

1

(Dliℓ1 + (1 − 2β )Dliℓ2 ) X˜ (t , β )dβ ∫ 1 ∫ 1 − X˜ T (t , β )Sd X˜ (t , β )dβ − X˜ T (t , β )dβ 0 0 } ∫ 1 X˜ (t , β )dβ , × hRd

(16) 1 S h

3.3. Derivative of discretized Lyapunov functional

{

ζ T (t)Ξlijℓ ζ (t) + 2ζ T (t)

0

if S > 0. Then substituting the relations (4) and (7) into the above inequality yields (16). ■

V˙ 2 (t) = 2e2ν t ζ T (t)AT

ρ¯ ijℓ (t)e

{

×

]

ˆ = R+ QNl ] , l ∈ 0, N0 , R

0

[

2ν t

∫

Proof. Using the same technique as in the proof of Gu, Kharitonov, and Chen (2003, Prop. 5.20), it can be easily shown that equality (15) holds, and the following inequality is true for t ≥ 0:

⎡

1 ∑ i,j,ℓ=0

× ζˆ (t , β )dβ, t ∈ [tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N, where Qˆ l = [Q0l Q1l . . . S = diag (S0 , S1 , . . . , SN ).

(19)

where Q˜ (t , η) = N0 ϱ(t)(Ql (η) − Ql−1 (η)) − ρ10 (t)Q˙ l (η) − ˜ ξ , η) = ρ11 (t)Q˙(l−1 (η), for t ∈ [)tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N, and R( ∂ R(ξ ,η) ∂ R(ξ ,η) eν (ξ +η) ∂ξ + ∂η . Furthermore, we can show the following.

Furthermore, if Sq > 0, q ∈ 0, N, then it also satisfies 1

eν (s−t) R(0, s − t)x(s)ds

t −τ

q=1

∑1

(18)

∫ eν (s−t −τ ) R(−τ , s − t)x(s)ds − xT (t − τ ) t −τ ] ∫ ∫ t 1 t T ˜ − x (s1 )R(s1 − t , s2 − t)ds1 x(s2 )ds2 2 t −τ t −τ [ + e2ν t xT (t)S(0)x(t) − e−2ντ xT (t − τ )S(−τ ) ] ∫ t 2ν (s−t) T ˙ × x(t − τ ) − e x (s)S(s − t)x(s)ds ,

h

∫

,

t −τ t

) 1 ( X˜ (t , β ) = col x(1) (t , β ), . . . , x(N) (t , β ) .

V (t) = e2ν t

} )

((

2ν +

N0 ln µl

) Pl−i +

σℓ

N0

σℓ

(Pl − Pl−1 )

)

+ Pl−i A0 + AT0 Pl−i + Q0,l−i + Q0T,l−i + S0 , Dli1ℓı = h(ν I + A0 )T Qı,l−i +

) N0 h ( Qıl − Qı,l−1 + hR0ı , σℓ

] 1[ Qıl = Q0l + (−1)ı−1 Q1l . . . QN −1,l + (−1)ı−1 QNl , 2 1[ Rmı = Rm0 + (−1)ı−1 Rm1 Rm1 + (−1)ı−1 Rm2 . . . 2 ] Rm,N −1 + (−1)ı−1 RmN , m ∈ {0, N }.

(20)

W.-H. Chen et al. / Automatica 86 (2017) 129–137

Proof. For t ∈ [tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N, with piecewise linear Ql as in (11), we have t

∫

∗

t −τ N

=

= 0

ρ1i (t)Ql−i (θq + β h)x(q) (t , β )dβ

[ Ξlijℓ ∗ ∗

i=0

1 ∑

1

∫

∑

0

q=1

ρ1i (t)h Q1,l−i − (1 − 2β )Q2,l−i X˜ (t , β )dβ.

(

)

(21)

i=0

Similarly, with piecewise linear R as in (13), we obtain t

∫

e

∫

ν (s−t)

R(0, s − t)x(s)ds

h (R01 − (1 − 2β )R02 ) X˜ (t , β )dβ,

(22)

0 t

∫

e

ν (s−t −τ )

R(−τ , s − t)x(s)ds

= e−ντ

1

h (RN1 − (1 − 2β )RN2 ) X˜ (t , β )dβ.

(23)

0

Note that for p, q ∈ 1, N and for (ξ , η) ∈ (θp , θp−1 ) × (θq , θq−1 ), the piecewise linear properties (11)–(13) imply 1

t

Dliℓ1 −Sd − hRd

∗

> 0, i, j ∈ 0, 1, l ∈ 1, N0 , Dliℓ2 0 −3Sd

(27)

] < 0, i, j, ℓ ∈ 0, 1, l ∈ 1, N0 ,

(28)

where h = τ /N, Qˆ l , R, S , Ξlijℓ , Dliℓı , Sd , and Rd are defined in Propositions 2–3, in which QqN0 are selected as (29)

Proof. In view of (16) and (27), there exists a scalar ϵ0 > 0 such that V (t) ≥ ϵ0 e2ν t ∥x(t)∥2 , t ≥ 0.

(30)

V˙ (t) < 0, t ∈ (tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N.

(31)

Then, recalling the fact that condition (29) implies the relation (10), and V (t) is continuous inside (tk−1 , tk ), k ∈ N, we can conclude from (10), (30) and (31) that It follows from (15) that there exists a positive scalar ϵ1 such that

∥x(t)∥2 ≤ (ϵ1 /ϵ0 )e−2ν t ∥φ∥2τ , t ≥ 0, which is the same as

∥x(t)∥ ≤

Then it follows that for t ∈ [tk,l−1 , tk,l ), l ∈ 1, N0 , k ∈ N,

∫

S

∥x(t)∥2 ≤ (1/ϵ0 )e−2ν t V (0), t ≥ 0.

(Qq−1,l − Qq,l ), l ∈ 0, N0 , h ˙ η) = 1 (Sq−1 − Sq ), S( h ) ∂ R(ξ , η) ∂ R(ξ , η) 1( + = Rp−1,q−1 − Rpq . ∂ξ ∂η h

Q˙ l (η) =

h

On the other hand, using similar arguments as in Gu et al. (2003, Prop. 5.21), it can be shown that condition (28) guarantees

t −τ

∫

R+

]

QqN0 = (I + C0 )T Qq0 , q ∈ 0, N .

t −τ 1

=

ˆ l−i Q 1

1

1

∑∫

and Rpq = RTqp , p, q ∈ 0, N, such that (8) and the following matrix inequalities hold:

[ µ ˆ lj Pl−i

eν (s−t) Q (t , s − t)x(s)ds

133

√ ϵ1 /ϵ0 e−ν t ∥φ∥τ , t ≥ 0,

Therefore, system (2) is UGES over S(σ1 , σ2 ) with exponential convergence rate ν . This completes the proof. ■

eν (s−t) Q˜ (t , s − t)x(s)ds

t −τ 1

∫

Remark 1. In Chen and Zheng (2009a), in order to obtain a delaydependent stability condition, the impulse-type ∫ t Newton–Leibnitz formula is used to treat the delayed term: t −τ x˙ (s)ds = x(t) −

[ N0 ϱ(t)h Q1l − Q1,l−1 − (1 − 2β )(Q2l − Q2,l−1 )

(

=

)

0 1 ∑

−

∑m′ (t)

] ρ1i (t)Q2,l−1 X˜ (t , β )dβ,

(24)

i=0

∫

t

∫

t −τ 1

˙ − t)x(s)ds e2ν (s−t) xT (s)S(s X˜ T (t , β )Sd X˜ (t , β )dβ,

=

(25)

0

∫

t

∫

t −τ 1

∫ ∫

= 0

t

˜ 1 − t , s2 − t)ds1 x(s2 )ds2 xT (s1 )R(s t −τ 1

X˜ T (t , α )hRd X˜ (t , β )dα dβ.

(26)

0

Thus, substituting (21)–(26) into (18)–(19) and using (4) and (7) yield (20). This completes the proof. ■ 3.4. Exponential stability criterion Based on Propositions 1–3, we have the following stability theorem for system (2). Theorem 1. Given scalar ν > 0, the zero solution of system (2) is UGES over S(σ1 , σ2 ) with exponential convergence rate ν , if for prescribed positive integers N0 , N, and scalars µl > 0, l ∈ 1, N0 , there exist n × n matrices Pl > 0, l ∈ 0, N0 , Sq > 0, Qq¯l , ¯l ∈ 0, N0 − 1,

x(t − τ ) − i=1 ∆x(ti′ ), where m′ (t) is the impulse number during the delay interval (t − τ , t), and ti′ , i ∈ 1, m′ (t), denote the impulse instants inside (t − τ , t). The delayed jump terms ∆x(ti′ ) are difficult to be handled via Lyapunov functional. Moreover, the indefiniteness of the number m′ (t) usually leads to the overdesign in bounding the delayed jump terms. The impulse-time-dependent discretized Lyapunov functional method proposed herein allows to avoid these technical obstacles, and thus can efficiently reduce the conservatism entailed in the Razumikhin-type method. As a tradeoff, the number of decision variables used to test the stability 2 is 2n (n + 1)(N0 + 2N + 3) + n2 (N + 1)(2N0 + N) + N0 . So large partition numbers N and N0 would increase the computational cost. Remark 2. When σ1 is given, for fixed partition numbers N and N0 , selecting appropriate µl , l ∈ 1, N0 , could yield a smaller σ0 . The following procedure is suggested for tuning µl , l ∈ 1, N0 . Step 1. Set σ0 = σ1 , and µl = µ, l ∈ 1, N0 . Then apply a 1-D search over µ > 0 to find an initial value of µ. Step 2. Adjust the value of µl , l ∈ 1, N0 by gradually decreasing the value of σ0 until the corresponding set of linear matrix inequalities (LMIs) is infeasible. Corollary 1. The zero solution of system (2) is UGES over S(σ1 , σ2 ), if for prescribed positive integers N0 and N, and scalars µl > 0, l ∈ 1, N0 , there exist n × n matrices Pl > 0, l ∈ 1, N0 , Sq , Qq¯l , ¯l ∈ 0, N0 − 1, and Rpq = RTqp , p, q ∈ 0, N, such that (8), (27), and (28) with ν = 0 and QqN0 defined in (29), are feasible.

134

W.-H. Chen et al. / Automatica 86 (2017) 129–137

Proof. If the matrix inequalities of (8), (27), and (28) with ν = 0 are feasible, then they are feasible for small enough ν > 0. Thus, by Theorem 1, the zero solution of system (2) is UGES over S(σ1 , σ2 ). ■ Corollary 2. For any given τ > 0, the zero solution of system (2) is UGES over S(σ1 , σ2 ), if for prescribed positive integer N0 , and scalars µl > 0, l ∈ 1, N0 , there exist n × n matrices Pl > 0, l ∈ 0, N0 , and S > 0, such that (8), and the following LMIs are feasible:

Ξ˜ lijℓ =

[ 1 Ξ˜ lijℓ ∗

] µ ˆ lj Pl−i A1 < 0, −S

i, j, ℓ ∈ 0, 1, l ∈ 1, N0 ,

Ξlij1 ℓ

where ˜

=µ ˆ lj

( N0 ln µl σℓ

Pl−i +

N0

σℓ

(Pl − Pl−1 ) + Pl−i A0 +

(32) AT0 Pl−i

)

+ S.

Proof. Suppose that there exist n × n matrices Pl > 0, S > 0, and positive scalars µl , l ∈ 1, N0 , such that the matrix inequalities (8) and (32) are satisfied. For any given delay τ > 0, choose N = 1, Qql = Rpq = 0, p, q ∈ 0, 1, l ∈ 0, N0 , S0 = S, and S1 = (1 − ϵ )S with ϵ ∈ (0, 1). Then, for small enough ϵ > 0, the matrix inequalities of (8) and (28), in which N = 1 and ν = 0, are also satisfied. Thus, in view of Corollary 1, we conclude that the zero solution of system (2) is UGES over S(σ1 , σ2 ). ■ Remark 3. Different from Corollary 1, the stability condition of Corollary 2 is independent of delay τ . Such delay-independent stability condition can also be derived by applying the impulse-timeLyapunov functional: V¯ (t , xt ) = ϕ (t)xT (t)P(t)x(t) + ∫dependent t T x (s)Sx(s)ds. When N0 = 1, this Lyapunov functional reduces t −τ to the one introduced in Chen et al. (2013). When N0 ≥ 2, this functional involves the partition on the impulse intervals. As will be shown in Example 1, the corresponding stability condition becomes less conservative as N0 increases. Remark 4. Recently, Lyapunov-based sufficient conditions for asymptotic stability of hybrid systems with memory were established in Liu and Teel (2016b) using generalized solutions (Liu & Teel, 2016a, 2016b). It is worth mentioning that some generalized solutions to hybrid systems do not have direct correspondence to the classical solutions to impulsive systems because such generalized solutions to hybrid systems have multiple jumps at the same time instant. In Liu and Teel (2016a, b), the stability of the scalar impulsive delay differential equation x˙ (t) = ax(t) + bx(t − τ ), t ̸ = kδ x(tk ) = ρ x(tk− ), t = kδ, k ∈ N

(33)

was studied in the framework of hybrid systems with memory via the Lyapunov functional method. The derived stability condition can be formulated as follows: there exist scalars µ > 0 and σ , such that ρ 2 ≤ e−σ δ , and the following matrix inequalities hold: (2a − σ )e−σ δi + µ be−σ δi

[

be−σ δi

−µ

]

< 0, i = 1, 2,

where δ1 = 0, and δ2 = δ . It is noted that the above condition can be reproduced from Corollary 2 when N0 = 1, P1 = P0 = 1, S = µ, and µ1 = e−σ δ .

4. L2 -gain criterion In this section, with the aid of the Lyapunov functional V (t), we can establish the following L2 -gain result.

Theorem 2. Given scalar γ > 0, system (2) is internally UGES and has L2 -gain less than γ over S(σ1 , σ2 ), if for prescribed positive integers N0 , N, and scalars µl > 0, l ∈ 1, N0 , there exist n × n matrices Pl > 0, l ∈ 1, N0 , Sq , Qq¯l , ¯l ∈ 0, N0 − 1, and Rpq = RTqp , p, q ∈ 0, N, such that (8), (27), and the following LMIs are satisfied:

[ ΞLlijℓ ∗ ∗

DLliℓ1 −Sd − hRd

∗

DLliℓ2 0 −3Sd

] < 0,

(34)

where i, j, ℓ ∈ 0, 1, l ∈ 1, N0 ,

ΞLlijℓ =

ˆ lj Pl−i H1 I1T µ + E˜ T E˜ − γ 2 I2T I2 , 0

]

[ Ξ0lijℓ ∗ [

]

D0liℓı , ı ∈ 1, 2. (−1)ı−1 hH1T Qı,l−i

DLliℓı =

In the above, I1 = [I 0], I2 = [0 0 I ], E˜ = [E 0 H2 ], Ξ0lijℓ and D0liℓı denote the ones corresponding to Ξlijℓ and Dliℓı , respectively, in which ν is chosen to be 0, Qˆ l , R, S , Ξlijℓ , Dliℓı Sd , and Rd are defined in Propositions 2–3, and QqN0 is defined in (29). Proof. Let V˜ (t) be the Lyapunov functional represented by (14) with ν = 0. Conditions (8) and (29) assure that V˜ (t) is nonnegative and decreasing at the impulse instants. Then, using the standard procedure, we can conclude that system (2) is internally UGES and has L2 -gain less than γ over S(σ1 , σ2 ). ■ Corollary 3. Given γ > 0, system (2) is internally UGES and has L2 gain less than γ over S(σ1 , σ2 ) for any τ ≥ 0, if for prescribed positive integer N0 , and scalars µl > 0, l ∈ 1, N0 , there exist n × n matrices Pl > 0, l ∈ 0, N0 , S > 0, and positive scalars αlijℓ , i, j, ℓ ∈ 0, 1, l ∈ 1, N0 , such that (8), and the following LMIs are satisfied for all i, j, ℓ ∈ 0, 1, l ∈ 1, N0 :

[ Ξ˜ lijℓ ∗

]

I1T µ ˆ lj Pl−i H1 + E˜ T E˜ − γ 2 I2T I2 < 0, 0

˜ are defined in ˜ lijℓ is defined in Corollary 2, and I1 , I2 , and E, where Ξ Theorem 2. The proof of Corollary 3 is similar to that of Corollary 2 and is thus omitted here. 5. Numerical examples In this section, the usefulness of the results derived in the preceding section will be illustrated by three numerical examples. Example 1. Consider a nondelayed linear impulsive system which is described by (2) with A1 = 0, and

[ A0 =

−1 0

0.1 0.2 , C0 = 1.2 0

]

[

]

0 . −0.5

Note that both the continuous and the discrete dynamics of the system are unstable. We distinguish two cases to determine the allowable ranged dwell-time such that the exponential stability can be guaranteed. Case 1: Periodic impulses case, i.e., {tk } ∈ S(σ , σ ). We apply Corollary 2 to estimate σmin and σmax for different N0 . By solving LMIs (8) and (32) with tuning the parameters µl , l ∈ 1, N0 , the obtained results are listed in Table 1. The analytical bounds are also listed there for comparison. It can be seen from Table 1 that the analytical bounds are achieved for N0 = 3. The same numerical result was also obtained in Briat (2013) and Briat and Seuret (2012b) by the sum-of-squares (SOS)based looped-functional approach. However, using the LMI-based

W.-H. Chen et al. / Automatica 86 (2017) 129–137

135

Fig. 1. The state responses of system (37) in Example 3: (a) without impulsive control; (b) with impulsive control. Table 1 Estimates of σmax and σmin for Case 1 of Example 1.

Table 3 Estimates of τmax for system (35)–(36) with different α of Example 2.

N0

1

2

3

Analytical

α

0.99

1.02

1.1

1.4

1.7

2

σmax σmin

0.4614 0.1826

0.5728 0.1824

0.5776 0.1824

0.5776 0.1824

σ = 0.2 σ = 0.5 σ = 0.7

8.890 6.876 6.618

4.079 4.983 5.223

2.038 3.024 3.390

0.504 1.383 1.613

N/A 0.804 1.081

N/A 0.430 0.739

Table 2 Estimates of σ0 min for Case 2 of Example 1. N0

50

60

80

σ0 min

0.1906 200

0.1881 240

0.187 320

NDV

looped-functional approach in Briat and Seuret (2012a), the derived σmin and σmax are 0.1824 and 0.576, respectively. Case 2: Aperiodic impulses case. Choose σ1 = 0.5776, and we seek a lower bound of σ0 such that the aperiodic impulsive system is uniformly exponentially stable over S(σ0 , σ1 ). The derived σ0 min from Corollary 2 for different N0 is presented in Table 2, where the acronym NDV means the number of decision variables. When N0 = 80, the execution time is about 1.5 s. It was reported in Briat (2013) that the allowable ranged dwell-time obtained from the SOS-based looped-functional approach is S(0.1824, 0.5776), and only 149 variables are involved. This shows that for nondelayed linear impulsive systems, the SOSbased looped-functional of Briat (2013) can provide better results than the proposed discretization LMI approach. How to establish stability criteria based SOS (Briat, 2016; Papachristodoulou, Peet, & Niculescu, 2007) for impulsive delay systems is a very meaningful topic which will be investigated in our future work. Example 2. Consider the following linear time-delay system

[ x˙ (t) =

−2 0

]

[

0 −1 x(t) + −0.9 −1

]

0 x(t − τ ). −1

(35)

System (35) is exponentially stable if and only if τ ≤ 6.17258, see Example 5.10 in Gu et al. (2003). Now we consider the case that this system is subjected to periodically impulsive perturbation with the following form x(tk ) = α x(tk− ), tk − tk−1 = σ , k ∈ N,

(36)

where α > 0. In what follows, for given three impulse periods: σ = 0.2, σ = 0.5, and σ = 0.7, we discuss how the sizes of α and τ affect the stability. Applying Corollary 2 with N0 = 2 to the three cases: σ = 0.2, 0.5, and 0.7, the obtained upper bound of α such that system (35)–(36) is exponentially stable for any τ ≥ 0 is αmax = 0.98, 0.95, and 0.931, respectively. This means that when α >

αmax , the stability may not be guaranteed for all τ ≥ 0. Thus, we need to resort to Corollary 1 for determining the upper bound of τ . For several given α > αmax , the maximum allowable values of τ obtained from Corollary 1 with N0 = N = 2 are listed in Table 3. The NDV is 65, and the execution time is about 15 s. Table 3 shows the relation among the impulse magnitude α , the impulse period σ , and the maximum allowable delay τmax given by Corollary 1. It should be pointed out that Corollary 1 is much less conservative than Theorem 1 in Chen and Zheng (2009a). In fact, when α ≥ 1.4, the conditions of Theorem 1 in Chen and Zheng (2009a) cannot be satisfied for any σ ≤ 0.7 and any τ > 0; when α = 1.02, for σ = 0.2, 0.5, and 0.7 successively, the maximum obtainable τmax from Theorem 1 in Chen and Zheng (2009a) is τmax = 0.935, 0.939, and 0.94, respectively; when α = 1.1, the corresponding maximum obtainable τmax for σ = 0.2, 0.5, and 0.7, is τmax = 0.51, 0.697, and 0.721, respectively. Example 3. Consider the linear time-delay system:

[ x˙ (t) =

0 0

]

[

0 −1 x(t) + 1 0

] [ ] −1 1 x(t − 1) + w (t), −0.9 1

z(t) = [0 1]x(t) + 0.1w (t)

(37)

The numerical simulation reports that the trajectories of system (37) are divergent, as shown in Fig. 1(a). In the following, we apply the reduced order impulsive controller:

∆x2 (t) = −0.5x2 (t − ), t = tk , k ∈ N,

(38)

to stabilize system (37). We assume that {tk } ∈ S(0.1, 0.2). Applying Theorem 2 with the choice of N0 = N = 2, µ1 = 0.82, and µ2 = 0.87, it has been found that the smallest value of γ for which the system (37)–(38) is stabilizable with L2 -gain γ is γ = 0.91. Because system (37) is unstable, and the impulses (38) are not the stabilizing-type as described in Chen and Zheng (2009b), the finite L2 -gain criteria given in Chen and Zheng (2009b) fail to work. For simulation studies, let w (t) =

{

4 sin(0.2t), 0,

0≤t≤5 , t>5

and

take the initial condition as φ (θ ) = (1, −1) for θ ∈ [−1, 0]. Let the impulsive instant tk be generated randomly with the constraint T

136

W.-H. Chen et al. / Automatica 86 (2017) 129–137

of 0.1 ≤ tk − tk−1 ≤ 0.2 for all k ∈ N. The state response is illustrated in Fig. 1(b). It can be seen that all state variables converge to zero when the disturbance disappears, indicating the stabilizing effect of the reduced order impulsive controller (38) on the state variables.

6. Conclusion In this paper, aiming at establishing delay-dependent criteria for exponential stability and L2 -gain of impulsive delay systems, we have proposed an impulse-time-dependent discretized Lyapunov functional method based on the impulse-type complete Lyapunov functional and the discretization procedure of Gu (1997). The proposed method is advantageous in that it is able to make good use of the information on the interaction among continuous dynamics, discrete dynamics, the size of time-delay, and the length of impulse intervals. The delay-dependent sufficient conditions for exponential stability and L2 -gain have been obtained in terms of LMIs. By setting some parameter matrices to be special values, these delay-dependent conditions can lead to new delayindependent conditions. The new delay-independent conditions are less conservative than the previous ones. Finally, we have presented three numerical examples which validate the effectiveness of the new theoretical results. Our further research will consider developing a discretization method for impulsive systems with time-varying delays.

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Wu-Hua Chen received the B.Sc. degree in Mathematics from Hubei Normal University, Huangshi, China, the M.Sc. degree in Basic Mathematics from Guangxi Normal University, Guilin, China, and the Ph.D. degree in Control Theory and Control Engineering from Huazhong University of Science and Technology, Wuhan, China, in 1988, 1991, and 2004, respectively. From 1991 to 2001, he was with Guangxi University for Nationalities, Nanning, China. In September 2001, he joined Guangxi University, Nanning, China, where he is currently a Professor. During April 2005 to October 2005, July 2007 to January 2008, and September 2008 to July 2009, he was a visiting fellow in University of Western Sydney, Australia. From January 2017 to February 2017, he was a temporary research scientist in the Science Program at Texas A&M University at Qatar. His research interests include time-delay systems, impulsive systems, switched systems, and reaction– diffusion systems.

Zhen Ruan was born in Nanning, Guangxi, China in 1992. She received the B.Sc. degree in Mathematics from Hubei Normal University, Huangshi, Hubei, China. Currently she is pursuing the M.Sc. degree at the College of Mathematics and Information Science, Guangxi University, Nanning, China. Her current research interests include impulsive control systems and switched control systems.

W.-H. Chen et al. / Automatica 86 (2017) 129–137 Wei Xing Zheng received the B.Sc. degree in Applied Mathematics in 1982, the M.Sc. degree in Electrical Engineering in 1984, and the Ph.D. in Electrical Engineering in 1989, all from Southeast University, Nanjing, China. He is currently a Professor at Western Sydney University, Sydney, Australia. Over the years he has also held various faculty/research/visiting positions at Southeast University, China; Imperial College of Science, Technology and Medicine, UK; University of Western Australia; Curtin University of Technology, Australia; Munich University of Technology, Germany; University of Virginia, USA; and University of California-Davis, USA. His research interests are in the areas of systems and controls, signal processing, and communications.

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Prof. Zheng is a Fellow of IEEE. Previously, he served as an Associate Editor for IEEE Transactions on Circuits and Systems-I: Fundamental Theory and Applications, IEEE Transactions on Automatic Control, IEEE Signal Processing Letters, IEEE Transactions on Circuits and Systems-II: Express Briefs, and IEEE Transactions on Fuzzy Systems, and as a Guest Editor for IEEE Transactions on Circuits and Systems-I: Regular Papers. Currently, he is an Associate Editor for Automatica, IEEE Transactions on Automatic Control (the second term), IEEE Transactions on Cybernetics, IEEE Transactions on Neural Networks and Learning Systems, IEEE Transactions on Control of Network Systems, and other scholarly journals. He is also an Associate Editor of IEEE Control Systems Society’s Conference Editorial Board. He has served as Publication Co-Chair of the 56th IEEE Conference on Decision and Control in Melbourne, Australia, December 2017.