Observer-based sliding mode control for discrete nonlinear systems with packet losses: An eventtriggered method

In this paper, the observer-based output feedback sliding mode control (SMC) problem is investigated for discrete delayed nonlinear systems subject to packet losses under the event-triggered strategy.

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Observer-based sliding mode control for discrete nonlinear systems with packet losses: an event-triggered method

Xinyu Guan, Jun Hu, Yunfei Cui & Long Xu

To cite this article: Xinyu Guan, Jun Hu, Yunfei Cui & Long Xu (2020) Observer-based sliding mode control for discrete nonlinear systems with packet losses: an event-triggered method, Systems Science & Control Engineering, 8:1, 175-188, DOI: 10.1080/21642583.2020.1734986

To link to this article: https://doi.org/10.1080/21642583.2020.1734986

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2020, VOL 8, NO 1, 175–188

https://doi.org/10.1080/21642583.2020.1734986

Observer-based sliding mode control for discrete nonlinear systems with packet losses: an event-triggered method

Xinyu Guana,b, Jun Hu a,b,c, Yunfei Cuia,band Long Xua,b

a Department of Mathematics, Harbin University of Science and Technology, Harbin, People’s Republic of China;bHeilongjiang Provincial Key Laboratory of Optimization Control and Intelligent Analysis for Complex Systems, Harbin University of Science and Technology, Harbin, People’s Republic of China;cSchool of Engineering, University of South Wales, Pontypridd, UK

ABSTRACT

In this paper, the observer-based output feedback sliding mode control (SMC) problem is

investi-gated for discrete delayed nonlinear systems subject to packet losses under the event-triggered

strategy It is assumed that the packet losses may occur in the control channel from the sensor to

the observer A suitable compensation strategy via the Bernoulli distributed random variable is used

to reduce the effects of packet losses In order to avoid the phenomenon of network congestion

dur-ing the networked transmission, an event-triggered mechanism is introduced to determine if the last

released measurement needs to be updated Based on the zero-order-hold (ZOH) measurement, an

output feedback observer is designed to reconstruct the system state This method can facilitate

the design of the discrete-time sliding surface A sufficient condition is proposed to guarantee the

stochastic stability of sliding mode dynamics systems by using linear matrix inequality (LMI) method,

and a novel observer-based sliding mode controller is synthesized to force the trajectories of the

error systems onto the pre-designed sliding mode surface within finite time Finally, an example is

given to illustrate the validity of the proposed theoretical result.

ARTICLE HISTORY

Received 2 January 2020 Accepted 23 February 2020

KEYWORDS

Event-triggered scheme; sliding mode control; packet losses; state observer; active compensation

1 Introduction

The sliding mode control (SMC) is an effective

con-trol technique, which has been widely discussed in the

control theory (Kchaou & EI-Hajjaji, 2017; Zhang, Shi,

& Xia,2010) The main idea of SMC is to select a suitable

sliding surface and design a discontinuous SMC law to

drive the system trajectories onto pre-designed sliding

surface, which can keep that the state trajectories stay in

the sliding surface thereafter (Cui, Hu, Wu, & Yang,2019;

Zhang, Hu, Liu, & Zhang, 2018; Zhang, Hu, Zhang,

& Chen,2020) The SMC has some great advantages

com-pared with the conventional control methods such as the

insensitivity the matched parameter variations and

exter-nal disturbances (Zhang et al.,2018) Therefore, the SMC

scheme has been widely used in the engineering fields,

such as robot manipulators, aircrafts, electrical motors

and so on (Tong, Lin, Huo, Jin, & Miao,2020) Considerable

research efforts have been devoted to the SMC problems

for various systems, for example, fuzzy systems (Zhang

et al., 2010), uncertain systems (Zhang & Xia, 2010),

Markov jump systems (Chen, Guo, & Ma,2019),

stochas-tic systems (Liu, Wu, Wu, Luo, & Franquelo,2019), and

discrete-time systems (That & Ha, 2015) Note that the

CONTACT Jun Hu jhu@hrbust.edu.cn, hujun2013@gmail.com

existence of the time delay would degrade the perfor-mance (Fei, Guan, & Gao,2018; Fei, Shi, Wang, & Wu,2018) Recently, in Chen et al (2019), the SMC problem has been investigated for a class of uncertain discrete delayed sys-tems with unmatched external disturbances and commu-nication constraints

In the practical applications, the data transmission is periodic with sampling and transmission at a fixed time interval in the networked environment (Hu, Wang, Liu, Zhang, & Navaratne, 2020) Therefore, a huge sample data needed to be calculated and transmitted How-ever, it is worth mentioning that the successive trans-missions inevitably lead to unnecessary space occupancy and energy waste Therefore, there is a need to pro-vide an effective method to determine whether the sampled signals should be sent out or not, which is commonly determined by certain criterion and guaran-tees the satisfactory performance (Dong, Wang, Shen,

& Ding,2016; Hu, Liu, Zhang, & Liu,2020; Zhang, Hu, Liu,

Yu, & Liu,2019) Due to the above situation, much effort has been devoted to present the proper communication protocols (Chu & Li,2019; Kumari, Bandyopadhyay, Kim,

& Shim,2019) Recently, the event-triggered mechanism

© 2020 The Author(s) Published by Informa UK Limited, trading as Taylor & Francis Group.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( http://creativecommons.org/licenses/by/4.0/ ), which permits unrestricted use,

has been introduced and some related works with the

event-triggered scheme have been given (Lu, Hu, Guo,

& Zhou,2018; Wu, Gao, Liu, & Li, 2017; Yao, Zhang, Li,

& Li, 2019) For example, based on the observer-based

control and state-feedback control scheme, the

event-triggered control problem of Markov jump systems (MJSs)

has been studied in Yao et al (2019) In Song, Wang,

and Niu (2019), the token-dependent SMC law has been

proposed, which can force the trajectory of error

sys-tems onto the designed sliding mode surface and ensure

that the estimation error system is asymptotically

sta-ble In Lu et al (2018), the multi-delay stochastic NCS

has been discussed and the event-triggered scheme has

been proposed by using the free-weighting matrix (FWM)

method and the integral inequality method Recently,

in Wu et al (2017), the event-triggered SMC method

has successfully applied to multi-loop control by taking

the limitation of shared communication into account

However, there are no results available on analysing

the observer-based SMC for discrete nonlinear systems

with the consideration of the event-triggered

mecha-nism, which motivates us to cope with this challenging

and meaningful topic

On another research front, the packet losses and

uncertain observations have been stirred much

atten-tion in the study of communicaatten-tion network (Dong, Hou,

Wang, & Ren,2018; Dong, Wang, Ding, & Gao,2016; Hu,

Wang, Liu, & Zhang,2019; Tan & Liu, 2012, 2013; Tan,

Liu, & Duan, 2012; Tan, Liu, & Shi, 2015; Tan, Yin, Liu,

Huang, & Zhao, 2018) Generally, the packet losses are

modelled by the Bernoulli distribution and the Markov

chain (Hu, Zhang, Kao, Liu, & Chen,2019; Hu, Zhang, Yu,

Liu, & Chen,2019; Wang, Dong, Shen, & Gao,2013)

Con-sider the phenomenon of packet losses, which may occur

in a feedback loop of the communication network, the

discrete-time integral sliding mode surfaces have been

designed via the packet losses probability and the

slid-ing mode controllers have been designed for network

control systems with continuous Markov packet losses

in Niu and Ho (2010) and Song, Chen, and Yam (2017),

respectively In Xue, Yu, and Wang (2019), the H∞

con-trol problem has been studied for discrete-time linear

time-delay systems with random packet losses and

quan-tization Besides, the sector-bounded method has been

applied to convert the quantitative control problem of

networked systems into a robust control problem with

uncertainty Two different schemes for the uncertain

lin-ear systems involving packet losses have been

consid-ered, they are the hold-input method and zero-input

method (Yang, Wang, Niu, & Li,2010), respectively It is

devoted to the problem of robust output-feedback SMC

for the networked systems involving both measuring

and actuation consecutive data packet losses In Argha,

Li, Su, and Nguyen (2016), a discrete-time SMC prob-lem with robust output feedback and packet losses has been studied However, it should be noted that there is a need to propose an event-based SMC scheme for discrete networked systems with packet losses and time-varying delays in order to fit the communication constraints Inspired by the above discussions, the main goal of this paper is to solve the observer-based SMC problem for a delayed system with event-triggered scheme and packet losses Here, the time-varying delays with known lower and upper bounds are considered Moreover, the packet losses are addressed by utilizing a Bernoulli distributed random variable Then, an observer-based sliding mode control method is given to fulfil the addressed problem The addressed problem has two challenges/difficulties

as follows: (1) How to deal with the effects of non-linear disturbances, time-varying delays and parameter-uncertainty on the discrete-time system simultaneously? (2) How to propose an efficient control method to atten-uate the effects from the phenomena of event-triggered mechanism and packet losses onto the whole control per-formance? In summary, we adopt the following solutions Firstly, we handle the parameter uncertainties and non-linear disturbances by using the norm-bounded conation and Lipschitz method, which are addressed by utilizing the linear matrix inequality technique Moreover, to tackle the effects of the event-triggered mechanism and packet losses, the trigger condition and compensator are pro-posed, respectively Based on the Lyapunov stability the-ory, the stochastic stability criterion is established for the addressed discrete delayed system Specifically, the main contributions of this paper are listed as follows (1) Both the event-triggered mechanism and packet losses are, for the first time, introduced together for the SMC prob-lem in order to reflect a more realistic environment; (2) A observer-based SMC method is given to compensate the effects of time-delay, packet losses and event-triggered mechanism; and (3) New sufficient condition is given to ensure the stochastic stability of resulted sliding mode dynamics and the reachability is shown

2 Problem formulation and preliminaries

In this section, the brief problem formulation is given and some useful lemmas are introduced

2.1 System model

In this paper, the concerned discrete delayed nonlinear system is described by

x k+1 = (A + A)x k + A d x k−τ k + B(u k + f(x k )),

y k = Cx k,

x k = φ k ∀ k ∈ [−τ M, 0], (1)

where x k∈Rn , u k∈Rm and y k∈Rp are the state

vec-tor, the control input and the output, respectively A, A d , B

and C are known constant matrices of appropriate

dimen-sions The parameter-uncertainty matrixA is assumed

to be norm-bounded of the following form:

where F is an unknown matrix satisfying FTF ≤ I, E and

H are known constant matrices of appropriate

dimen-sions The nonlinear disturbance f (x k ) with known bound

satisfies

whereλ > 0 represents a known constant.

Assumption 2.1: The positive integerτ k describes the

discrete time-varying delay and satisfies

τ m ≤ τ k ≤ τ M, withτ mandτ Mbeing the bounds

2.2 Packet losses

It is assumed that the packet losses will occur In order to

compensate the packet losses, the following method will

be utilized in this paper:

y ck =



y k, if data packet is received,

y k−1, if data packet is lost.

(4)

Equation (4) describes that y ck is equal to y kwhen the

packet is perfectly received at time k, otherwise y ck is

equal to y k−1when a packet loss occurs The probability of

packet losses is determined by the Bernoulli distribution

as follows:

Pr{θ k = 1} = ¯θ, Pr{θ k = 0} = 1 − ¯θ, (5)

where 0≤ ¯θ < 1 stands for the probability Equation (4)

can be rewritten in the following form:

y ck = (1 − θ k )y k + θ k y k−1. (6)

2.3 Event-triggered scheme

In this paper, the event detector is introduced to reduce

the network burdens and then save the limited

commu-nication resources In particular, an event-triggered

sam-pling strategy is used to determine whether or not the

current measurement output y k should be transmitted

When the data is transmitted from observer to controller,

the following event-triggered condition is introduced to reduce the utilization of the network resources

eTˆx k e ˆx k ≥ δˆxT

where 0< δ < 1 is a constant, and e ˆx k = ˆx k − ˆx ¯k with

ˆx k and ˆx ¯k being the current measurement and the last released one, respectively  > 0 is known weighting

matrix

To end this section, the following lemmas are intro-duced to facilitate further derivations

Lemma 2.1: For any real vectors a, b and matrix P > 0 of appropriate dimensions, we have

aTb + bTa ≤ aTPa + bTP−1b.

Lemma 2.2: Let Q = QT, N and H be real matrices of

appro-priate dimensions For any F satisfying F = FT≤ I, Q +

NFH + HTFTNT< 0 if and only if there exists a scalar ε > 0 such that Q + εNNT+ ε−1HTH or equivalently

⎡

T

⎤

⎦ < 0.

Lemma 2.3 (Schur complement lemma): Given

con-stant matrices S1,S2,S3, where S1=ST

1and 0 < S2=ST

2,

then S1+ST

3S2−1S3< 0 if and only if



S1 ST 3

∗ −S2



< 0 or



−S2 S3

∗ S1



< 0.

3 Observer-based sliding mode control

In this section, we aim to establish the event-triggered SMC scheme for considered discrete-time networked sys-tem with packet losses Firstly, an estimator is constructed

to estimate the unmeasurable state variables In addition, the observer-based controller is designed to force the state trajectories onto the pre-designed sliding surface The detailed flowchart is given in Figure1

Remark 3.1: As illustrated in Figure1, it is easy to see that the signal transmitted is divided into the following steps

Step 1: the original measurement output y k is obtained

at the time instant k Step 2: the phenomenon of packet

losses is described when transmitting the signal, where a random variable obeying the Bernoulli distribution is

uti-lized Therefore, the original measurement y kis replaced

by the updated signal y ck Step 3: a state observer is con-structed in order to obtain the unmeasurable state vari-able Step 4: the event-triggered is introduced to decrease

the network burdens, and the state variable x ¯kis

transmit-ted to plant at the trigger instant ¯k.

Figure 1.Event-triggered control with packet losses.

3.1 Estimator design

Firstly, the following state estimator is constructed:

ˆx k+1 = Aˆx k + Bu k + L[y ck − (1 − ¯θ)ˆy k],

whereˆx k ∈Rn denotes the estimator state and L∈Rn×p

is estimator gain to be determined later

The sliding mode surface is constructed as follows:

where the matrix G∈Rm ×n will be designed later to

ensure the non-singularity of GB.

According to SMC theory, when the system

trajecto-ries reach the sliding mode surface, the ideal condition

satisfies S k+1= S k= 0 Therefore, we have

S k+1= Gˆx k+1

= G[Aˆx k + Bu k + L(y ck − (1 − ¯θ)ˆy k )]

The equivalent control law is then obtained as

u k eq = −(GB)−1G[A ˆx k + L(y ck − (1 − ¯θ)ˆy k )]. (11)

By the triggered condition in (7), the

event-triggered equivalent control law can be rewritten as

fol-lows:

u ¯k

eq = −(GB)−1G[A ˆx ¯k + L(y ck − (1 − ¯θ)ˆy k )]. (12)

Substituting (12) into (8) yields

ˆx k+1 = [A − (θ k − ¯θ)LC](I − ¯G)ˆx k + ¯GAe ˆx k

+ (1 − θ k )(I − ¯G)LCe k + θ k (I − ¯G)

× LCe k−1+ θ k (I − ¯G)LCˆx k−1, (13)

where ¯G = B(GB)−1G Hence, based on (1) and (8), the

error dynamic can be derived as

e k+1 = [A + A − (1 − θ k )LC]e k + A d e k−τ k

+ Bf(x k ) − θ k LCe k−1 − θ k LC ˆx k−1

+ [A + (θ k − ¯θ)LC]ˆx k + A d ˆx k−τ k (14)

3.2 Stability analysis

In this subsection, a sufficient condition is given to ensure the stochastic stability of the sliding mode dynamics based on the linear matrix inequality technique

Theorem 3.1: Consider the sliding mode surface (9) Given

scalars ε1> 0 and ε2> 0, then the resulting closed-loop systems composed of (13) and (14) are stochastically stable,

if there exist symmetric matrices P1> 0, P2> 0, Q2> 0 and

W > 0 satisfying

⎡

⎣ ∗11 1222 1323

∗ ∗ −ε1I

⎤

⎦ < 0, (15)

BTP1B < ε2I, (16)

where

 ,

11=

⎡

⎢

⎣

⎤

⎥

⎦ ,

⎡

⎢

⎣

⎤

⎥

23=

 1

23 2 23

 ,

22=

⎡

⎢

⎣

∗ 66 AT

d P1A d

⎤

⎥

⎦ ,

 , 11= 111 112 131

 ,

12=



1

 ,

1

11=

⎡

⎣0 3αC

3ATP1B

⎤

⎦ ,

1

12=

⎡

⎣

√

5αCTWTB ¯θATP1B 0 0

⎤

⎦ ,

1

13=

⎡

α1CTWT α2CTWTB √

3αCTWT

⎤

⎦ ,

1

21=

⎡

⎢

⎣

√

6 ¯θCTWT CTWT √

3 ¯θCTWTB

⎤

⎥

⎦ ,

1

22=

⎡

⎢

⎣

√

3αCTWT 0

2 ¯θCTWT

√

2 ¯θCTWTB √

3αCTWT

⎤

⎥

⎦ ,

13=

⎡

⎢

⎢

⎢

⎢

⎢

⎣

(1 − ¯θ)CTWTE 0

− ¯θCTWTE − ¯θCTWTE

− ¯θCTWTE − ¯θCTWTE

ATd P1E ATd P1E

AT

d P1E

⎤

⎥

⎥

⎥

⎥

⎥

⎦ ,

1

23=

⎡

⎢

⎢

⎢

⎢

⎢

⎣

√

3P1E 0

0 (2 − ¯θ)P1E

⎤

⎥

⎥

⎥

⎥

⎥

⎦ ,

2

23=



0 0 0 0 0 0 0 0 0 0 −ε1I

T

,

α1=√5(1 − ¯θ), α2=√2(1 − ¯θ), α3= 2 − ¯θ,

11= −P1+ (τ M − τ m + 2)P2+ 2ATP1A

+ (6 + ¯θ)ε2λ2I + δ + ε1HTH,

13= (1 − ¯θ)ATWC,

14 15= ¯θATWC,

34 35= − ¯θATWC,

36 37= ATP1A d − (1 − ¯θ)CTWTA d,

46= − ¯θCTWTA d,

66= 2AT

d P1A d − Q2,

77= 2AT

d P1A d − P2,

22= −diag{P1, P1, BTP1B, BTP1B, ¯ θBTP1B,

BTP1B, P1,(1 − ¯θ)P1,(1 − ¯θ)BTP1B, P1,

¯θP1, P1, ¯θBTP1B, P1, ¯θP1, ¯θBTP1B, P1},

33= (6 + ¯θ)ε2λ2I + (τ M − τ m + 1)Q2− P1+ P2

+ ε1HTH.

Here, G = BTP1and L = P−11 W is the observer gain.

Proof: Select the following Lyapunov–Krasovskii

func-tional:

V k=

5

p=1

V k p,

V k1= ˆxT

k P1ˆx k,

V k2= eT

k P1e k,

V k3=

r−1

j=r−τ k

ˆxT

j P2ˆx j+

−τ m

j=−τ M+1

r−1

i=r+j

ˆxT

i P2ˆx i,

V k4=

g−1

l =g−τ k

eTl Q2e l+

−τ m

l =−τ M+1

g−1

t =l+g

eTt Q2e t,

V k5= eT

k−1 P2e k−1 + ˆxT

k−1 P2ˆx k−1.

DefiningE{V1

k} =E{V1

k+1} −E{V1

k}, then we have

E{V1

k } = {[A − (θ k − ¯θ)LC](I − ¯G)ˆx k + ¯GAe ˆx k

+ (1 − θ k )(I − ¯G)LCe k

+ θ k (I − ¯G)LCe k−1 + θ k (I − ¯G)LC

ˆx k−1}TP1{[A − (θ k − ¯θ)LC](I − ¯G)ˆx k

+ ¯GAe ˆx + (1 − θ k )(I − ¯G)LCe k

+ θ k (I − ¯G)LCe k−1 + θ k (I − ¯G)LC

ˆx k−1} − ˆxT

k P1ˆx k

= ˆxT

k AT(I − ¯G)TP1(I − ¯G)Aˆx k

+ 2ˆxT

k AT(I − ¯G)TP1¯GAe ˆx k + 2(1 − ¯θ)ˆxT

k AT(I − ¯G)TP1(I − ¯G)LCe k

+ 2 ¯θˆxT

k AT(I − ¯G)TP1(I − ¯G)LCe k−1

+ 2 ¯θˆxT

k AT(I − ¯G)TP1(I − ¯G)LCˆx k−1

+ α2ˆxT

k CTLT(I − ¯G)TP1(I − ¯G)LCˆx k

+ 2α2ˆxT

k CTLT(I − ¯G)TP1(I − ¯G)LCe k

− 2α2ˆxT

k CTLT(I − ¯G)TP1(I − ¯G)LCe k−1

− 2α2ˆxT

k CTLT(I − ¯G)TP1(I − ¯G)LCˆx k−1

+ eT

ˆx k AT¯GTP1¯GAe ˆx k + 2(1 − ¯θ)eT

ˆx k AT¯GTP1(I − ¯G)LCe k

+ 2 ¯θeT

ˆx k AT¯GTP1(I − ¯G)LCe k−1

+ 2 ¯θeT

ˆx k AT¯GTP1(I − ¯G)LCˆx k−1

+ (1 − ¯θ)eT

k CTLT(I − ¯G)TP1(I − ¯G)LCe k

+ ¯θeT

k−1CTLT(I − ¯G)TP1(I − ¯G)LCe k−1

+ 2 ¯θeT

k−1CTLT(I − ¯G)TP1(I − ¯G)LCˆx k−1

+ ¯θˆxT

k−1 CTLT(I − ¯G)TP1(I − ¯G)LCˆx k−1

− ˆxT

where ¯G = B(GB)−1G, E{(θ k − ¯θ)2} = (1 − ¯θ) ¯θ = α2,

E{(θ k − ¯θ)(1 − θ k )} = −α2andE{θ k − ¯θ} = 0 Next, it is

easy to obtain that

E{V1

k } ≤ 2ˆxT

k ATP1A ˆx k + 2ˆxT

k ATGT(GB)−1GA ˆx k

+ 2(1 − ¯θ)ˆxT

k ATP1LCe k

− 2(1 − ¯θ)ˆxT

k ATGT(GB)−1GLCe k

+ 2 ¯θˆxT

k ATP1LCe k−1

− 2 ¯θˆxT

k ATGT(GB)−1GLCe k−1

+ 2 ¯θˆxT

k ATP1LC ˆx k−1

− 2 ¯θˆxT

k ATGT(GB)−1GLC ˆx k−1

+ 2α2ˆxT

k CTLTP1LC ˆx k

+ 2α2ˆxT

k CTLTGT(GB)−1GLC ˆx k

+ 2α2ˆxT

k CTLTP1LCe k

− 2α2ˆxT

k CTLTGT(GB)−1GLCe

k

− 2α2ˆxT

k CTLTP1LCe k−1

+ 2α2ˆxT

k CTLTGT(GB)−1GLCe k−1

− 2α2ˆxT

k CTLTP1LC ˆx k−1

+ 2α2ˆxT

k CTLTGT(GB)−1GLC ˆx k−1

+ eT

ˆx k ATGT(GB)−1GAe ˆx k

+ 2(1 − ¯θ)eT

k CTLTP1LCe k

+ 2(1 − ¯θ)eT

k CTLTGT(GB)−1GLCe k

+ 2 ¯θeT

k−1 CTLTP1LCe k−1

+ 2 ¯θeT

k−1 CTLTGT(GB)−1GLCe k−1

+ 2 ¯θeT

k−1CTLTP1LC ˆx k−1

− 2 ¯θeT

k−1CTLTGT(GB)−1GLC ˆx k−1

+ 2 ¯θˆxT

k−1 CTLTP1LC ˆx k−1

+ 2 ¯θˆxT

k−1 CTLTGT(GB)−1GLC ˆx k−1

− ˆxT

By applying Lemma 2.1, we can get

− 2(1 − ¯θ)ˆxT

k ATGT(GB)−1GLCe k

≤ (1 − ¯θ)ˆxT

k ATGT(GB)−1GA ˆx k

+ (1 − ¯θ)eT

− 2 ¯θˆxT

k ATGT(GB)−1GLCe k−1

≤ ¯θˆxT

k ATGT(GB)−1GA ˆx k

+ ¯θeT

− 2 ¯θˆxT

k ATGT(GB)−1GLC ˆx k−1

≤ ¯θˆxT

k ATGT(GB)−1GA ˆx k

+ ¯θˆxT

− 2α2ˆxT

k CTLTGT(GB)−1GLCe k

≤ α2ˆxT

k CTLTGT(GB)−1GLC ˆx k

+ α2eTk CTLTP1LCe k, (22)

2α2ˆxT

k CTLTGT(GB)−1GLCe k−1

≤ α2ˆxT

k CTLTGT(GB)−1GLC ˆx k

+ α2eTk−1CTLTP1LCe k−1, (23)

2α2ˆxT

k CTLTGT(GB)−1GLC ˆx k−1

≤ α2ˆxT

k CTLTGT(GB)−1GLC ˆx k

+ α2ˆxT

− 2 ¯θeT

k−1CTLTGT(GB)−1GLC ˆx k−1

≤ ¯θeT

k−1CTLTGT(GB)−1GLCe

k−1

+ ¯θˆxT

2α2ˆxT

k CTLTP1LCe k

≤ α2ˆxT

k CTLTP1LC ˆx k + α2eTk CTLTP1LCe k, (26)

− 2α2ˆxT

k CTLTP1LCe k−1

≤ α2ˆxT

k CTLTP1LC ˆx k

+ α2eTk−1CTLTP1LCe k−1, (27)

− 2α2ˆxT

k CTLTP1LC ˆx k−1

≤ α2ˆxT

k CTLTP1LC ˆx k

+ α2ˆxT

2 ¯θeT

k−1 CTLTP1LC ˆx k−1

≤ ¯θeT

k−1CTLTP1LCe k−1

+ ¯θˆxT

Substituting (19) and (29) into (18) and taking the

mathe-matical expectation, one has

E{V1

k } ≤ ˆxT

k {2ATP1A + 5α2CTLTP1LC

+ (3 + ¯θ)ATGT(GB)−1GA + 5α2CT

× LTGT(GB)−1GLC }ˆx k + eT

k {3(1 − ¯θ)

× CTLTP1LC + 2(1 − ¯θ)

× CTLTGT(GB)−1GLC + 2α2

× CTLTP1LC }e k + eT

k−1{4 ¯θCTLTP1LC

+ 3 ¯θCTLTGT(GB)−1GLC

+ 2α2CTLTP1LC }e k−1 + ˆxT

k−1 {5 ¯θCTLT

× P1LC + 2 ¯θCTLTGT(GB)−1GLC

+ 2α2CTLTP1LC }ˆx k−1

+ 2(1 − ¯θ)ˆxT

k ATP1LCe k

+ 2 ¯θˆxT

k ATP1LCe k−1

+ 2 ¯θˆxT

k ATP1LC ˆx k−1

+ eT

ˆx k ATGT(GB)−1GAe ˆx k

− ˆxT

Besides,

E{V2

k } = {¯Ae k + A d e k−τ k + Bf(x k ) − θ k LCe k−1

− θ k LC ˆx k−1 + [A + (θ k − ¯θ)LC]ˆx k

+ A d ˆx k−τ k}TP1{¯Ae k + A d e k−τ k

+ Bf(x k ) − θ k LCe k−1 − θ k LC ˆx k−1

+ [A + (θ k − ¯θ)LC]ˆx k + A d ˆx k −τ k}

− eT

k P1e k

=E{eT

k ¯ATP1¯Ae k + 2eT

k ¯ATP1A d e k−τ k

+ 2eT

k ¯ATP1Bf (x k ) − 2 ¯θeT

k ¯ATP1LCe k−1

− 2 ¯θeT

k ¯ATP1LC ˆx k−1 + 2eT

k ¯ATP1[A

+ (θ k − ¯θ)LC]ˆx k + 2eT

k ¯ATP1A d ˆx k−τ k

+ eT

k−τ k ATd P1A d e k−τ k

+ 2eT

k−τ k ATd P1Bf (x k )

− 2 ¯θeT

k−τ k ATd P1LCe k−1

− 2 ¯θeT

k−τ k ATd P1LC ˆx k−1

+ 2eT

k−τ k ATd P1Aˆx k

+ 2eT

k−τ k ATd P1A d ˆx k−τ k

+ fT(x k )BTP1Bf (x k )

− 2 ¯θfT(x k )BTP1LCe k−1

− 2 ¯θfT(x k )BTP1LC ˆx k−1 + 2fT(x k )BTP1Aˆx k

+ 2fT(x k )BTP1A d ˆx k−τ k

+ ¯θeT

k−1 CTLTP1LCe k−1

+ 2 ¯θeT

k−1 CTLTP1LC ˆx k−1

− 2 ¯θeT

k−1CTLTP1Aˆx k

− 2 ¯θeT

k−1CTLTP1A d ˆx k −τ k

− 2α2eTk−1 CTLTP1LC ˆx k

+ ¯θˆxT

k−1 CTLTP1LC ˆx k−1

− 2 ¯θˆxT

k−1 CTLTP1Aˆx k

− 2 ¯θˆxT

k−1CTLTP1A d ˆx k −τ k

− 2α2ˆxT

k−1CTLTP1LC ˆx k

+ ˆxT

k ATP1Aˆx k

+ 2ˆxT

k ATP1A d ˆx k−τ k

+ α2ˆxT

k CTLTP1LC ˆx k

+ ˆxT

k −τ k ATd P1A d ˆx k −τ k } − eT

k P1e k, (31)

where ¯A = A + A − (1 − θ k )LC Then, we obtain

E{V2

k } = eT

k (A + A)TP1(A + A)e k

− 2(1 − ¯θ)eT

k (A + A)TP1LCe k

+ 2(1 − ¯θ)eT

k CTLTP1LCe k

+ 2eT

k (A + A)TP1A d e k−τ k

− 2(1 − ¯θ)eT

k CTLTP1A d e k −τ k

+ 2eT

k (A + A)TP1Bf (x k )

− 2(1 − ¯θ)eT

k CTLTP1Bf (x k )

− 2 ¯θeT

k (A + A)TP1LCe k−1

− 2 ¯θeT

k (A + A)TP1LC ˆx k−1

+ 2eT

k (A + A)TP1Aˆx k

− 2(1 − ¯θ)eT

k CTLTP1Aˆx k

+ 2α2eTk CTLTP1LC ˆx k

+ 2eT

k (A + A)TP1Aˆx k

− 2(1 − ¯θ)eT

k CTLTP1Aˆx k

+ eT

k−τ k ATd P1A d e k−τ k

+ 2eT

k−τ k ATd P1Bf (x k )

− 2 ¯θeT

k−τ k ATd P1LCe k−1

− 2 ¯θeT

k−τ k ATd P1LC ˆx k−1

+ 2eT

k−τ k ATd P1Aˆx k

+ 2eT

k−τ k ATd P1A d ˆx k−τ k

+ fT(x k )BTP1Bf (x k )

− 2 ¯θfT(x k )BTP1LCe k−1

− 2 ¯θfT(x k )BTP1LC ˆx k−1

+ 2fT(x k )BTP1Aˆx k

+ 2fT(x k )BTP1A d ˆx k −τ k

+ ¯θeT

k−1 CTLTP1LCe k−1

+ 2 ¯θeT

k−1 CTLTP1LC ˆx k−1

− 2 ¯θeT

k−1 CTLTP1Aˆx k

− 2α2eTk−1CTLTP1LC ˆx k

− 2 ¯θeT

k−1CTLTP1A d ˆx k−τ k

+ ¯θˆxT

k−1 CTLTP1LC ˆx k−1

− 2 ¯θˆxT

k−1 CTLTP1Aˆx k

− 2α2ˆxT

k−1CTLTP1LC ˆx k

− 2 ¯θˆxT

k−1CTLTP1A d ˆx k −τ k

+ ˆxT

k ATP1Aˆx k

+ 2ˆxT

k ATP1A d ˆx k−τ k

+ ˆxT

k−τ k ATd P1A d ˆx k −τ k

+ α2ˆxT

k CTLTP1LC ˆx k

− eT

By applying Lemma 2.1, it follows that

2α2eTk CTLTP1LCLC ˆx k

≤ α2eTk CTLTP1LCe k

+ α2ˆxT

2eTk −τ k ATd P1Bf (x k )

≤ eT

k −τ k ATd P1A d e k −τ k

+ fT(x k )BTP1Bf (x k ), (34)

− 2 ¯θfT(x k )BTP1LCe k−1

≤ ¯θfT(x k )BTP1Bf (x k )

+ ¯θeT

− 2 ¯θfT(x k )BTP1LC ˆx k−1

≤ ¯θfT(x k )BTP1Bf (x k )

+ ¯θˆxT

2fT(x k )BTP1Aˆx k

≤ fT(x k )BTP1Bf (x k ) + ˆxT

k ATP1Aˆx k, (37)

2fT(x k )BTP1A d ˆx k −τ k

≤ fT(x k )BTP1Bf (x k )

+ ˆxT

2 ¯θeT

k−1 CTLTP1LC ˆx k−1

≤ ¯θeT

k−1 CTLTP1LCe k−1

+ ¯θˆxT

2eTk (A + A)TP1Bf (x k )

≤ fT(x k )BTP1Bf (x k )

+ eT

k (A + A)TP1(A + A)e k, (40)

− 2(1 − ¯θ)eT

k CTLTP1Bf (x k )

≤ (1 − ¯θ)fT(x k )BTP1Bf (x k )

+ (1 − ¯θ)eT

2eTk (A + A)TP1Aˆx k

≤ eT

k (A + A)TP1(A + A)e k

+ ˆxT

− 2α2eTk−1CTLTP1LC ˆx k

≤ α2eTk−1 CTLTP1LCe k−1

+ α2ˆxT

− 2α2ˆxT

k−1CTLTP1LC ˆx k

≤ α2ˆxT

k CTLTP1LC ˆx k

+ α2ˆxT

− 2(1 − ¯θ)eT

k (A + A)TP1LCe k

≤ (1 − ¯θ)eT

k (A + A)TP1(A + A)e k

+ (1 − ¯θ)eT

According to (3), we have

fT(x k )BTP1Bf (x k )

≤ BTP1B λ2x k2≤ ε2λ2Ix kTx k

= ε2λ2I ˆxT

k ˆx k + ε2λ2IeTk e k (46)

Substituting (33)–(46) into (32), we obtain

E{V2

k } ≤ (4 − ¯θ)eT

k (A + A)TP1(A + A)e k

+ 2(1 − ¯θ)eT

k CTLTP1LCe k

+ 2eT

k (A + A)TP1A d e k−τ k

− 2(1 − ¯θ)eT

k CTLTP1A d e k−τ k

− 2 ¯θeT

k (A + A)TP1LCe k−1

− 2 ¯θeT

k (A + A)TP1LC ˆx k−1

− 2(1 − ¯θ)eT

k CTLTP1Aˆx k

+ 2eT

k (A + A)TP1A d ˆx k−τ k

− 2(1 − ¯θ)eT

k CTLTP1A d ˆx k−τ k + α2eTk−1CTLTP1LCe k−1

+ 2eT

k −τ k ATd P1A d e k−τ k

− 2 ¯θeT

k −τ k ATd P1LCe k−1

− 2 ¯θeT

k −τ k ATd P1LC ˆx k−1

+ 2eT

k −τ k ATd P1Aˆx k

+ 2eT

k −τ k ATd P1A d ˆx k −τ k

+ (6 + ¯θ)ε2λ2I ˆxT

k ˆx k

+ (6 + ¯θ)ε2λ2IeTk e k

+ (2 ¯θ + 1)eT

k−1 CTLTP1LCe k−1

− 2 ¯θeT

k−1 CTLTP1Aˆx k

+ α2eTk−1 CTLTP1LCe k−1

− 2 ¯θeT

k−1CTLTP1A d ˆx k −τ k

+ 3 ¯θˆxT

k−1CTLTP1LC ˆx k−1

− 2 ¯θˆxT

k−1 CTLTP1Aˆx k

− 2 ¯θˆxT

k−1 CTLTP1A d ˆx k−τ k

+ α2ˆxT

k−1CTLTP1LC ˆx k−1

+ 3ˆxT

k ATP1Aˆx k

+ 2ˆxT

k ATP1A d ˆx k−τ k

+ 2ˆxT

k−τ k ATd P1A d ˆx k−τ k

+ 4α2ˆxT

k CTLTP1LC ˆx k − eT

E{V3

k} =E

⎧

⎨

⎩

r

j=r+1−τ k+1

ˆxT

j P2ˆx j−

r−1

j=r−τ k

ˆxT

j P2ˆx j

+

−τ m

j=−τ M+1

⎛

⎝ r

i=r+j+1

−

r−1

i=r+j

⎞

⎠ ˆxT

i P2ˆx i

⎫

⎬

⎭

≤ (τ M − τ m + 1)ˆxT

k P2ˆx k

− ˆxT

E{V4

k} =E

⎧

⎨

⎩

g

l=g+1−τ k+1

eTl Q2e l−

g−1

l=g−τ k

eTl Q2e l

+ −τ

m

l=−τ M+1

⎛

⎝ g

t=l+1+g

−

g−1

t=l+g

⎞

⎠ eT

t Q2e t

⎫

⎬

⎭

≤ (τ M − τ m + 1)eT

k Q2e k

− eT

E{V5

k } = eT

k P2e k − eT

k−1P2e k−1+ ˆxT

k P2ˆx k

− ˆxT

In terms of the event-triggered condition (7), we can achieve

δˆxT

k ˆx k − 2δˆxT

k e ˆx k + δeT

ˆx k e ˆx k − eT

ˆx k e ˆx k ≥ 0 (51) Combining (30), (47) with (48)–(51), we have

E{V k } ≤ ξ(k)Tϒ1ξ(k),

where

ξ(k) =ˆxT

k eTˆx k eTk eTk−1 ˆxT

k−1 eTk−τ k ˆxT

k−τ k

T

,

ϒ1=

⎡

⎢

⎢

⎢

⎢

⎢

⎣

ϒ11 −δ ϒ13 ϒ14 ϒ15 ϒ16 ϒ16

∗ ∗ ϒ33 ϒ34 ϒ35 ϒ36 ϒ36

⎤

⎥

⎥

⎥

⎥

⎥

⎦ ,

ϒ11= 3ATP1A + 9α2CTLTP1LC − P1+ δ

+ (τ M − τ m + 2)P2+ 2ATP1A

+ 5α2CTLTGT(GB)−1GLC + (6 + ¯θ)ε2λ2I

+ (3 + ¯θ)ATGT(GB)−1GA,

ϒ13= (1 − ¯θ)ATP1LC − (1 − ¯θ)ATP1LC,

ϒ14= ϒ15= ¯θATP1LC − ¯θATP1LC,

ϒ16= ATP1A d,

ϒ22= δ −  + ATGT(GB)−1GA,