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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.
Systems Science & Control Engineering An Open Access Journal ISSN: (Print) 2164-2583 (Online) Journal homepage: https://www.tandfonline.com/loi/tssc20 Observer-based sliding mode control for discrete nonlinear systems with packet losses: an eventtriggered 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 © 2020 The Author(s) Published by Informa UK Limited, trading as Taylor & Francis Group Published online: 04 Mar 2020 Submit your article to this journal Article views: 266 View related articles View Crossmark data Full Terms & Conditions of access and use can be found at https://www.tandfonline.com/action/journalInformation?journalCode=tssc20 SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 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,b and Long Xua,b a Department of Mathematics, Harbin University of Science and Technology, Harbin, People’s Republic of China; b Heilongjiang Provincial Key Laboratory of Optimization Control and Intelligent Analysis for Complex Systems, Harbin University of Science and Technology, Harbin, People’s Republic of China; c School of Engineering, University of South Wales, Pontypridd, UK ABSTRACT ARTICLE HISTORY 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 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 during 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 Received January 2020 Accepted 23 February 2020 Introduction The sliding mode control (SMC) is an effective control 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 compared with the conventional control methods such as the insensitivity the matched parameter variations and external 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), stochastic systems (Liu, Wu, Wu, Luo, & Franquelo, 2019), and discrete-time systems (That & Ha, 2015) Note that the CONTACT Jun Hu KEYWORDS Event-triggered scheme; sliding mode control; packet losses; state observer; active compensation existence of the time delay would degrade the performance (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 systems with unmatched external disturbances and communication 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 However, it is worth mentioning that the successive transmissions inevitably lead to unnecessary space occupancy and energy waste Therefore, there is a need to provide an effective method to determine whether the sampled signals should be sent out or not, which is commonly determined by certain criterion and guarantees 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 jhu@hrbust.edu.cn, hujun2013@gmail.com © 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, distribution, and reproduction in any medium, provided the original work is properly cited 176 X GUAN ET AL 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 eventtriggered 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 systems onto the designed sliding mode surface and ensure that the estimation error system is asymptotically stable 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 mechanism, 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 attention in the study of communication 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) Consider 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 sliding 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∞ control problem has been studied for discrete-time linear time-delay systems with random packet losses and quantization 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 linear systems involving packet losses have been considered, 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 problem 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 nonlinear disturbances, time-varying delays and parameteruncertainty on the discrete-time system simultaneously? (2) How to propose an efficient control method to attenuate the effects from the phenomena of event-triggered mechanism and packet losses onto the whole control performance? In summary, we adopt the following solutions Firstly, we handle the parameter uncertainties and nonlinear 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 proposed, respectively Based on the Lyapunov stability theory, 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 problem 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 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 xk+1 = (A + yk = Cxk , A)xk + Ad xk−τk + B(uk + f (xk )), SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL xk = φk ∀ k ∈ [−τM , 0], (1) where xk ∈ Rn , uk ∈ Rm and yk ∈ Rp are the state vector, the control input and the output, respectively A, Ad , B and C are known constant matrices of appropriate dimensions The parameter-uncertainty matrix A is assumed to be norm-bounded of the following form: A = EFH, (2) where F is an unknown matrix satisfying F T F ≤ I, E and H are known constant matrices of appropriate dimensions The nonlinear disturbance f (xk ) with known bound satisfies f (xk ) ≤ λ xk , Assumption 2.1: The positive integer τk describes the discrete time-varying delay and satisfies τm ≤ τk ≤ τM , with τm and τM being 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: yck = yk , if data packet is received, yk−1 , if data packet is lost ¯ Pr{θk = 0} = − θ, k exˆk ≥ δ xˆ kT¯ xˆ k¯ , (7) where < δ < is a constant, and exˆk = xˆ k − xˆ k¯ with xˆ k and xˆ k¯ being the current measurement and the last released one, respectively > is known weighting matrix To end this section, the following lemmas are introduced to facilitate further derivations Lemma 2.1: For any real vectors a, b and matrix P > of appropriate dimensions, we have aT b + bT a ≤ aT Pa + bT P−1 b Lemma 2.2: Let Q = QT , N and H be real matrices of appropriate dimensions For any F satisfying F = F T ≤ I, Q + NFH + HT F T NT < if and only if there exists a scalar ε > such that Q + εNNT + ε−1 HT H or equivalently ⎤ ⎡ Q εN HT ⎣ ∗ −εI ⎦ < ∗ ∗ −εI Lemma 2.3 (Schur complement lemma): Given constant matrices S1 , S2 , S3 , where S1 = S1T and < S2 = S2T , then S1 + S3T S2−1 S3 < if and only if S1 ∗ S3T represents a known constant 177 (6) 2.3 Event-triggered scheme In this paper, the event detector is introduced to reduce the network burdens and then save the limited communication resources In particular, an event-triggered sampling strategy is used to determine whether or not the current measurement output yk should be transmitted When the data is transmitted from observer to controller, In this section, we aim to establish the event-triggered SMC scheme for considered discrete-time networked system 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 Figure Remark 3.1: As illustrated in Figure 1, it is easy to see that the signal transmitted is divided into the following steps Step 1: the original measurement output yk 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 utilized Therefore, the original measurement yk is replaced by the updated signal yck Step 3: a state observer is constructed in order to obtain the unmeasurable state variable Step 4: the event-triggered is introduced to decrease the network burdens, and the state variable xk¯ is transmit¯ ted to plant at the trigger instant k 178 X GUAN ET AL Figure Event-triggered control with packet losses where G¯ = B(GB)−1 G Hence, based on (1) and (8), the error dynamic can be derived as 3.1 Estimator design Firstly, the following state estimator is constructed: ¯ yk ], xˆ k+1 = Aˆxk + Buk + L[yck − (1 − θ)ˆ yˆ k = C xˆ k , ek+1 = [A + + Bf (xk ) − θk LCek−1 − θk LC xˆ k−1 (8) where xˆ 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: Sk = Gˆxk , (9) where the matrix G ∈ Rm×n will be designed later to ensure the non-singularity of GB According to SMC theory, when the system trajectories reach the sliding mode surface, the ideal condition satisfies Sk+1 = Sk = Therefore, we have Sk+1 = Gˆxk+1 ¯ yk )] = G[Aˆxk + Buk + L(yck − (1 − θ)ˆ = ¯ + [ A + (θk − θ)LC]ˆ xk + Ad xˆ k−τk 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 > 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 > and W > satisfying ⎡ The equivalent control law is then obtained as ¯ ukeq −1 = −(GB) ¯ yk )] G[Aˆxk¯ + L(yck − (1 − θ)ˆ 12 23 ⎦ 22 ∗ ⎤ 13 < 0, BT P1 B < ε2 I, where 11 = 11 12 ∗ 22 ⎡ˆ ¯ xk + GAe ¯ xˆ ¯ − G)ˆ xˆ k+1 = [A − (θk − θ)LC](I k ¯ ¯ + (1 − θk )(I − G)LCe k + θk (I − G) 11 (13) 11 ⎢ ∗ =⎢ ⎣ ∗ ∗ , −δ δ − ∗ ∗ (15) −ε1 I (12) Substituting (12) into (8) yields ¯ xˆ k−1 , × LCek−1 + θk (I − G)LC 11 ⎣ ∗ ∗ (11) By the event-triggered condition in (7), the eventtriggered equivalent control law can be rewritten as follows: (14) 3.2 Stability analysis (10) ¯ yk )] ukeq = −(GB)−1 G[Aˆxk + L(yck − (1 − θ)ˆ A − (1 − θk )LC]ek + Ad ek−τk ˆ 13 ˆ 33 ∗ ˆ 14 ⎤ ⎥ ⎥ ˆ 34 ⎦ , −P2 (16) SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL ⎡ 12 22 12 ⎢ =⎢ ⎣ ˆ 35 ⎡ −P2 ⎢ =⎣ ∗ ∗ = ¯ 12 = ¯ 12 ¯ 13 ¯ 21 ¯ 22 13 23 ˆ 46 ˆ 66 ˆ 46 ∗ ¯ 12 , ¯1 ¯1 22 23 = 23 23 , ⎤ ¯ 11 = 0 0 0 0 0 0 0 0 0 0 −ε1 I √ √ ¯ α2 = 2(1 − θ), ¯ α3 = − θ, ¯ α1 = 5(1 − θ), 23 = T , ˆ 11 = −P1 + (τM − τm + 2)P2 + 2AT P1 A ⎥ ATd P1 Ad ⎦ , ˆ 77 ¯ 11 21 ¯ 11 ⎤ 0 ⎥ ⎥ ˆ 37 ⎦ , ˆ 46 0 ˆ 36 ˆ 46 179 ¯ λ2 I + δ + (6 + θ)ε ¯1 11 ¯1 12 ¯1 13 , , √ T ⎤ 3A P1 B ⎦, 0 ⎡ 3αC T W T ⎣ = 0 0 ⎤ ⎡√ ¯ T P1 B 5αC T W T B θA 0 =⎣ 0 AT P1 B ⎦, 0 α3 AT P1 ⎡ ⎤ 0 ⎦ =⎣ √ 0T T , T T T T α1 C W α2 C W B 3αC W √ ⎡√ ¯ T T T T ¯ T W T B⎤ 6θC W C W 3θC ⎢ ⎥ 0 ⎥, =⎢ ⎣ ⎦ 0 0 0 ⎡√ 3αC T W T √ 0T T ⎢ ¯ W 2θC =⎢ ⎣ 0 0 ⎤ √ 0T T √ 0T T ¯ W B 2θC 3αC W ⎥ ⎥, ⎦ 0 0 ⎤ ⎡ 0 ⎥ ⎢ 0 ⎥ ⎢ ⎥ ⎢(1 − θ)C T T ¯ W E ⎥ ⎢ ⎢ ¯ TWTE ¯ TWTE ⎥ = ⎢ −θC −θC ⎥, ⎥ ⎢ T T T T ¯ ¯ ⎢ −θC W E −θC W E ⎥ ⎥ ⎢ ⎦ ⎣ ATd P1 E ATd P1 E ATd P1 E ATd P1 E ⎡√ ⎤ 3P1 E ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ =⎢ 0 ⎥, ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ¯ 1E (2 − θ)P + ε1 HT H, ˆ 13 = (1 − θ)A ¯ T WC, ˆ 14 = ˆ 15 = θA ¯ T WC, ¯ T WC, ˆ 34 = ˆ 35 = −θA ˆ 36 = ˆ 37 = AT P1 Ad − (1 − θ)C ¯ T W T Ad , ˆ 46 = −θ¯ C T W T Ad , ˆ 66 = 2AT P1 Ad − Q2 , d ˆ 77 = 2AT P1 Ad − P2 , d 22 ¯ T P1 B, = −diag{P1 , P1 , BT P1 B, BT P1 B, θB ¯ , (1 − θ)B ¯ T P1 B, P1 , BT P1 B, P1 , (1 − θ)P ¯ T P1 B, P1 , θP ¯ , θB ¯ T P1 B, P1 }, θ¯ P1 , P1 , θB ˆ 33 = (6 + θ)ε ¯ λ2 I + (τM − τm + 1)Q2 − P1 + P2 + ε1 HT H Here, G = BT P1 and L = P1−1 W is the observer gain Proof: Select the following Lyapunov–Krasovskii functional: p Vk = Vk , p=1 Vk1 = xˆ kT P1 xˆ k , Vk2 = eTk P1 ek , −τm r−1 xˆ iT P2 xˆ i , j=−τM +1 i=r+j j=r−τk −τm g−1 Vk4 eTl Q2 el = l=g−τk Vk5 = r−1 xˆ jT P2 xˆ j + Vk3 = eTk−1 P2 ek−1 g−1 eTt Q2 et , + l=−τM +1 t=l+g T + xˆ k−1 P2 xˆ k−1 } − E{V }, then we have Defining E{ Vk1 } = E{Vk+1 k ¯ xk + GAe ¯ xˆ ¯ E{ Vk1 } = {[A − (θk − θ)LC](I − G)ˆ k ¯ + (1 − θk )(I − G)LCe k ¯ ¯ + θk (I − G)LCe k−1 + θk (I − G)LC ¯ xk ¯ xˆ k−1 }T P1 {[A − (θk − θ)LC](I − G)ˆ ¯ xˆ + (1 − θk )(I − G)LCe ¯ + GAe k k 180 X GUAN ET AL ¯ ¯ + θk (I − G)LCe k−1 + θk (I − G)LC − 2α xˆ kT C T LT P1 LC xˆ k−1 xˆ k−1 } − xˆ kT P1 xˆ k + 2α xˆ kT C T LT GT (GB)−1 GLC xˆ k−1 ¯ T P1 (I − G)Aˆ ¯ xk = xˆ kT AT (I − G) + eTxˆ AT GT (GB)−1 GAexˆk k ¯ T P1 GAe ¯ xˆ + 2ˆxkT AT (I − G) k ¯ T C T LT P1 LCek + 2(1 − θ)e k ¯ T P1 (I − G)LCe ¯ ¯ x T AT (I − G) + 2(1 − θ)ˆ k k ¯ T C T LT GT (GB)−1 GLCek + 2(1 − θ)e k ¯ T P1 (I − G)LCe ¯ + 2θ¯ xˆ kT AT (I − G) k−1 ¯ T C T LT P1 LCek−1 + 2θe k−1 ¯ T P1 (I − G)LC ¯ + 2θ¯ xˆ kT AT (I − G) xˆ k−1 ¯ T C T LT GT (GB)−1 GLCek−1 + 2θe k−1 ¯ T P1 (I − G)LC ¯ + α xˆ kT C T LT (I − G) xˆ k ¯ T C T LT P1 LC xˆ k−1 + 2θe k−1 ¯ T P1 (I − G)LCe ¯ + 2α xˆ kT C T LT (I − G) k ¯ T C T LT GT (GB)−1 GLC xˆ k−1 − 2θe k−1 ¯ T P1 (I − G)LCe ¯ − 2α xˆ kT C T LT (I − G) k−1 T + 2θ¯ xˆ k−1 C T LT P1 LC xˆ k−1 ¯ T P1 (I − G)LC ¯ − 2α xˆ kT C T LT (I − G) xˆ k−1 T + 2θ¯ xˆ k−1 C T LT GT (GB)−1 GLC xˆ k−1 ¯ xˆ + eTxˆ AT G¯ T P1 GAe k − xˆ kT P1 xˆ k k ¯ ¯ T AT G¯ T P1 (I − G)LCe + 2(1 − θ)e k xˆ k By applying Lemma 2.1, we can get ¯ ¯ T AT G¯ T P1 (I − G)LCe + 2θe k−1 xˆ k ¯ x T AT GT (GB)−1 GLCek − 2(1 − θ)ˆ k ¯ ¯ T AT G¯ T P1 (I − G)LC + 2θe xˆ k−1 xˆ k ¯ x T AT GT (GB)−1 GAˆxk ≤ (1 − θ)ˆ k ¯ T P1 (I − G)LCe ¯ ¯ T C T LT (I − G) + (1 − θ)e k k ¯ T C T LT P1 LCek , + (1 − θ)e k ¯ T P1 (I − G)LCe ¯ ¯ T C T LT (I − G) + θe k−1 k−1 ≤ θ¯ xˆ kT AT GT (GB)−1 GAˆxk T ¯ T P1 (I − G)LC ¯ + θ¯ xˆ k−1 C T LT (I − G) xˆ k−1 (17) ¯ } = (1 − θ) ¯ θ¯ = α , where G¯ = B(GB)−1 G, E{(θk − θ) ¯ ¯ = Next, it is E{(θk − θ)(1 − θk )} = −α and E{θk − θ} easy to obtain that E{ Vk1 } ≤ 2ˆxkT AT P1 Aˆxk + 2ˆxkT AT GT (GB)−1 GAˆxk ¯ x T AT P1 LCek + 2(1 − θ)ˆ k ¯ x T AT GT (GB)−1 GLCek − 2(1 − θ)ˆ k + 2θ¯ xˆ kT AT P1 LCek−1 − 2θ¯ xˆ kT AT GT (GB)−1 GLCek−1 + 2θ¯ xˆ kT AT P1 LC xˆ k−1 − 2θ¯ xˆ kT AT GT (GB)−1 GLC xˆ k−1 + 2α xˆ kT C T LT P1 LC xˆ k + 2α xˆ kT C T LT GT (GB)−1 GLC xˆ k + 2α xˆ kT C T LT P1 LCek − 2α xˆ kT C T LT GT (GB)−1 GLCek − 2α xˆ kT C T LT P1 LCek−1 + 2α xˆ kT C T LT GT (GB)−1 GLCek−1 (19) − 2θ¯ xˆ kT AT GT (GB)−1 GLCek−1 ¯ T P1 (I − G)LC ¯ ¯ T C T LT (I − G) + 2θe xˆ k−1 k−1 − xˆ kT P1 xˆ k , (18) ¯ T C T LT P1 LCek−1 , + θe k−1 (20) − 2θ¯ xˆ kT AT GT (GB)−1 GLC xˆ k−1 ≤ θ¯ xˆ kT AT GT (GB)−1 GAˆxk T + θ¯ xˆ k−1 C T LT P1 LC xˆ k−1 , (21) − 2α xˆ kT C T LT GT (GB)−1 GLCek ≤ α xˆ kT C T LT GT (GB)−1 GLC xˆ k + α eTk C T LT P1 LCek , (22) 2α xˆ kT C T LT GT (GB)−1 GLCek−1 ≤ α xˆ kT C T LT GT (GB)−1 GLC xˆ k + α eTk−1 C T LT P1 LCek−1 , (23) 2α xˆ kT C T LT GT (GB)−1 GLC xˆ k−1 ≤ α xˆ kT C T LT GT (GB)−1 GLC xˆ k T + α xˆ k−1 C T LT P1 LC xˆ k−1 , (24) ¯ T C T LT GT (GB)−1 GLC xˆ k−1 − 2θe k−1 ≤ θ¯ eTk−1 C T LT GT (GB)−1 GLCek−1 T + θ¯ xˆ k−1 C T LT P1 LC xˆ k−1 , 2α xˆ kT C T LT P1 LCek (25) SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL ≤ α xˆ kT C T LT P1 LC xˆ k + α eTk C T LT P1 LCek , − 2θ¯ eTk A¯ T P1 LC xˆ k−1 + 2eTk A¯ T P1 [ A (26) ¯ xk + 2eTk A¯ T P1 Ad xˆ k−τk + (θk − θ)LC]ˆ − 2α xˆ kT C T LT P1 LCek−1 ≤ α xˆ kT C T LT P1 LC xˆ k + eTk−τk ATd P1 Ad ek−τk + α eTk−1 C T LT P1 LCek−1 , + 2eTk−τk ATd P1 Bf (xk ) (27) − 2θ¯ eTk−τk ATd P1 LCek−1 − 2α xˆ kT C T LT P1 LC xˆ k−1 ≤ α xˆ kT C T LT P1 LC xˆ k − 2θ¯ eTk−τk ATd P1 LC xˆ k−1 T + α xˆ k−1 C T LT P1 LC xˆ k−1 , (28) + 2eTk−τk ATd P1 Aˆxk 2θ¯ eTk−1 C T LT P1 LC xˆ k−1 ≤ + 2eTk−τk ATd P1 Ad xˆ k−τk ¯ T C T LT P1 LCek−1 θe k−1 + f T (xk )BT P1 Bf (xk ) T + θ¯ xˆ k−1 C T LT P1 LC xˆ k−1 (29) ¯ T (xk )BT P1 LCek−1 − 2θf Substituting (19) and (29) into (18) and taking the mathematical expectation, one has − 2θ¯ f T (xk )BT P1 LC xˆ k−1 E{ Vk1 } ≤ + 2f T (xk )BT P1 Aˆxk xˆ kT {2AT P1 A + 5α C T LT P1 LC + 2f T (xk )BT P1 Ad xˆ k−τk ¯ T GT (GB)−1 GA + 5α C T + (3 + θ)A + θ¯ eTk−1 C T LT P1 LCek−1 ¯ GLC}ˆxk + eTk {3(1 − θ) −1 T T × L G (GB) + 2θ¯ eTk−1 C T LT P1 LC xˆ k−1 ¯ × C L P1 LC + 2(1 − θ) T T T T T −1 × C L G (GB) T T ×C L 181 GLC + 2α − 2θ¯ eTk−1 C T LT P1 Aˆxk − 2θ¯ eTk−1 C T LT P1 Ad xˆ k−τk ¯ T LT P1 LC P1 LC}ek + eTk−1 {4θC − 2α eTk−1 C T LT P1 LC xˆ k ¯ T LT GT (GB)−1 GLC + 3θC T + θ¯ xˆ k−1 C T LT P1 LC xˆ k−1 T ¯ T LT + 2α C L P1 LC}ek−1 + xˆ k−1 {5θC T T ¯ L G (GB) × P1 LC + 2θC T T T −1 T − 2θ¯ xˆ k−1 C T LT P1 Aˆxk GLC T − 2θ¯ xˆ k−1 C T LT P1 Ad xˆ k−τk T T + 2α C L P1 LC}ˆxk−1 T − 2α xˆ k−1 C T LT P1 LC xˆ k ¯ x T AT P1 LCek + 2(1 − θ)ˆ k + xˆ kT AT P1 Aˆxk + 2θ¯ xˆ kT AT P1 LCek−1 + 2ˆxkT AT P1 Ad xˆ k−τk + 2θ¯ xˆ kT AT P1 LC xˆ k−1 + α xˆ kT C T LT P1 LC xˆ k + eTxˆ AT GT (GB)−1 GAexˆk k − xˆ kT P1 xˆ k Besides, ¯ k + Ad ek−τ + Bf (xk ) − θk LCek−1 E{ Vk2 } = {Ae k T + xˆ k−τ AT P1 Ad xˆ k−τk } − eTk P1 ek , k d (30) where A¯ = A + A − (1 − θk )LC Then, we obtain E{ Vk2 } = eTk (A + A)T P1 (A + A)ek ¯ − θk LC xˆ k−1 + [ A + (θk − θ)LC]ˆ xk ¯ T (A + − 2(1 − θ)e k ¯ k + Ad ek−τ + Ad xˆ k−τk }T P1 {Ae k ¯ T C T LT P1 LCek + 2(1 − θ)e k + Bf (xk ) − θk LCek−1 − θk LC xˆ k−1 + 2eTk (A + ¯ + [ A + (θk − θ)LC]ˆ xk + Ad xˆ k−τk } ¯ T C T LT P1 Ad ek−τ − 2(1 − θ)e k k − eTk P1 ek + 2eTk (A + ¯ k + 2eT A¯ T P1 Ad ek−τ = E{eTk A¯ T P1 Ae k k ¯ T A¯ T P1 LCek−1 + 2eTk A¯ T P1 Bf (xk ) − 2θe k A)T P1 LCek A)T P1 Ad ek−τk A)T P1 Bf (xk ) ¯ T C T LT P1 Bf (xk ) − 2(1 − θ)e k − 2θ¯ eTk (A + A)T P1 LCek−1 (31) 182 X GUAN ET AL ¯ T (A + − 2θe k + 2eTk (A + A)T P1 LC xˆ k−1 2eTk−τk ATd P1 Bf (xk ) A)T P1 Aˆxk ≤ eTk−τk ATd P1 Ad ek−τk ¯ T C T LT P1 Aˆxk − 2(1 − θ)e k + f T (xk )BT P1 Bf (xk ), + 2eTk (A + ¯ T (xk )BT P1 Bf (xk ) ≤ θf A)T P1 Aˆxk ¯ T C T LT P1 Aˆxk − 2(1 − θ)e k ¯ T C T LT P1 LCek−1 , + θe k−1 + eTk−τk ATd P1 Ad ek−τk − 2θ¯ f T (xk )BT P1 LC xˆ k−1 + 2eTk−τk ATd P1 Bf (xk ) ¯ T (xk )BT P1 Bf (xk ) ≤ θf ¯ T AT P1 LC xˆ k−1 − 2θe k−τk d T ≤ f T (xk )BT P1 Bf (xk ) + xˆ kT AT P1 Aˆxk , (37) 2f T (xk )BT P1 Ad xˆ k−τk + 2eTk−τk ATd P1 Ad xˆ k−τk ≤ f T (xk )BT P1 Bf (xk ) + f T (xk )BT P1 Bf (xk ) T + xˆ k−τ AT P1 Ad xˆ k−τk , k d ¯ T (xk )BT P1 LCek−1 − 2θf (38) 2θ¯ eTk−1 C T LT P1 LC xˆ k−1 ¯ T (xk )BT P1 LC xˆ k−1 − 2θf ¯ T C T LT P1 LCek−1 ≤ θe k−1 + 2f T (xk )BT P1 Aˆxk T + θ¯ xˆ k−1 C T LT P1 LC xˆ k−1 , + 2f T (xk )BT P1 Ad xˆ k−τk 2eTk (A + ¯ T C T LT P1 LCek−1 + θe k−1 (39) A)T P1 Bf (xk ) ≤ f T (xk )BT P1 Bf (xk ) ¯ T C T LT P1 LC xˆ k−1 + 2θe k−1 + eTk (A + ¯ T C T LT P1 Aˆxk − 2θe k−1 A)T P1 (A + A)ek , (40) ¯ T C T LT P1 Bf (xk ) − 2(1 − θ)e k − 2α eTk−1 C T LT P1 LC xˆ k ¯ T (xk )BT P1 Bf (xk ) ≤ (1 − θ)f ¯ T C T LT P1 Ad xˆ k−τ − 2θe k−1 k ¯ T C T LT P1 LCek , + (1 − θ)e k T + θ¯ xˆ k−1 C T LT P1 LC xˆ k−1 2eTk (A + T − 2θ¯ xˆ k−1 C T LT P1 Aˆxk + xˆ kT T − 2θ¯ xˆ k−1 C T LT P1 Ad xˆ k−τk (41) A)T P1 Aˆxk ≤ eTk (A + T − 2α xˆ k−1 C T LT P1 LC xˆ k A)T P1 (A + A)ek T A P1 Aˆxk , (42) − 2α eTk−1 C T LT P1 LC xˆ k + xˆ kT AT P1 Aˆxk + 2ˆxkT AT P1 Ad xˆ k−τk ≤ α eTk−1 C T LT P1 LCek−1 T + xˆ k−τ AT P1 Ad xˆ k−τk k d + α xˆ kT C T LT P1 LC xˆ k , (43) T − 2α xˆ k−1 C T LT P1 LC xˆ k + α xˆ kT C T LT P1 LC xˆ k (32) ≤ α xˆ kT C T LT P1 LC xˆ k T + α xˆ k−1 C T LT P1 LC xˆ k−1 , By applying Lemma 2.1, it follows that ¯ T (A + − 2(1 − θ)e k 2α eTk C T LT P1 LCLC xˆ k ¯ T (A + ≤ (1 − θ)e k α eTk C T LT P1 LCek + α xˆ kT C T LT P1 LC xˆ k , (36) T 2f (xk )B P1 Aˆxk + 2eTk−τk ATd P1 Aˆxk ≤ (35) T C T LT P1 LC xˆ k−1 , + θ¯ xˆ k−1 − 2θ¯ eTk−τk ATd P1 LCek−1 − eTk P1 ek (34) ¯ T (xk )BT P1 LCek−1 − 2θf + 2α eTk C T LT P1 LC xˆ k A)T P1 LCek A)T P1 (A + ¯ T C T LT P1 LCek + (1 − θ)e k (33) (44) A)ek (45) SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL According to (3), we have E{ Vk3 } = E ≤ B P1 Bλ xk r r−1 xˆ jT P2 xˆ j − ⎩ j=r+1−τk+1 f T (xk )BT P1 Bf (xk ) T ⎧ ⎨ ≤ = ε2 λ2 IˆxkT xˆ k + ε2 λ2 IeTk ek j=−τM +1 (46) r ⎝ + xˆ jT P2 xˆ j j=r−τk ⎛ −τm ε2 λ2 IxkT xk 183 ⎞ r−1 ⎠ xˆ iT P2 xˆ i − i=r+j+1 i=r+j ⎫ ⎬ ⎭ ≤ (τM − τm + 1)ˆxkT P2 xˆ k Substituting (33)–(46) into (32), we obtain ¯ T (A + E{ Vk2 } ≤ (4 − θ)e k A)T P1 (A + T − xˆ k−τ P2 xˆ k−τk , k ⎧ g ⎨ E{ Vk } = E eTl Q2 el − ⎩ A)ek ¯ T C T LT P1 LCek + 2(1 − θ)e k + 2eTk (A + −τm + ¯ T C T LT P1 Ad ek−τ − 2(1 − θ)e k k A)T P1 LCek−1 ¯ T (A + − 2θe k A)T P1 LC xˆ k−1 l=−τM +1 eTl Q2 el ⎛ g l=g−τk g−1 ⎝ ⎠ eTt Q2 et − t=l+1+g ⎞ t=l+g ⎭ − eTk−τk Q2 ek−τk , (49) E{ Vk5 } = eTk P2 ek − eTk−1 P2 ek−1 + xˆ kT P2 xˆ k A)T P1 Ad xˆ k−τk T − xˆ k−1 P2 xˆ k−1 ¯ T C T LT P1 Ad xˆ k−τ − 2(1 − θ)e k k (50) In terms of the event-triggered condition (7), we can achieve + α eTk−1 C T LT P1 LCek−1 + 2eTk−τk ATd P1 Ad ek−τk δ xˆ kT xˆ k − 2δ xˆ kT exˆk + δeTxˆ k ¯ T AT P1 LCek−1 − 2θe k−τk d exˆk − eTxˆ k exˆk ≥ (51) Combining (30), (47) with (48)–(51), we have ¯ T AT P1 LC xˆ k−1 − 2θe k−τk d E{ Vk } ≤ ξ(k)T ϒ1 ξ(k), + 2eTk−τk ATd P1 Aˆxk where + 2eTk−τk ATd P1 Ad xˆ k−τk ¯ T C T LT P1 Ad xˆ k−τ − 2θe k−1 k T ξ(k) = xˆ kT eTxˆ eTk eTk−1 xˆ k−1 k ⎡ ϒ11 −δ ϒ13 ϒ14 ⎢ ∗ ϒ 0 22 ⎢ ⎢ ∗ ∗ ϒ ϒ ⎢ 33 34 ⎢ ϒ1 = ⎢ ∗ ∗ ∗ ϒ44 ⎢ ⎢ ∗ ∗ ∗ ∗ ⎢ ⎣ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ T + 3θ¯ xˆ k−1 C T LT P1 LC xˆ k−1 ϒ11 = AT P1 A + 9α C T LT P1 LC − P1 + δ ¯ λ2 Iˆx T xˆ k + (6 + θ)ε k ¯ λ2 IeT ek + (6 + θ)ε k + (2θ¯ + 1)eTk−1 C T LT P1 LCek−1 ¯ T C T LT P1 Aˆxk − 2θe k−1 + α eTk−1 C T LT P1 LCek−1 eTk−τk ϒ15 ϒ35 ϒ55 ∗ ∗ T xˆ k−τ k ϒ16 ϒ36 ϒ46 ϒ46 ϒ66 ∗ T − 2θ¯ xˆ k−1 C T LT P1 Aˆxk + (τM − τm + 2)P2 + 2AT P1 A T − 2θ¯ xˆ k−1 C T LT P1 Ad xˆ k−τk ¯ λ2 I + 5α C T LT GT (GB)−1 GLC + (6 + θ)ε T + α xˆ k−1 C T LT P1 LC xˆ k−1 ¯ T GT (GB)−1 GA, + (3 + θ)A + 3ˆxkT AT P1 Aˆxk ¯ T P1 LC − (1 − θ) ¯ AT P1 LC, ϒ13 = (1 − θ)A + 2ˆxkT AT P1 Ad xˆ k−τk ¯ T P1 LC − θ¯ AT P1 LC, ϒ14 = ϒ15 = θA T + 2ˆxk−τ AT P1 Ad xˆ k−τk k d ϒ16 = + 4α xˆ kT C T LT P1 LC xˆ k − eTk P1 ek , ⎫ ⎬ ≤ (τM − τm + 1)eTk Q2 ek ¯ T C T LT P1 Aˆxk − 2(1 − θ)e k + 2eTk (A + g−1 l=g+1−τk+1 A)T P1 Ad ek−τk ¯ T (A + − 2θe k (48) (47) ϒ22 = δ AT P1 Ad , − + AT GT (GB)−1 GA, T , ⎤ ϒ16 ⎥ ⎥ ϒ36 ⎥ ⎥ ⎥ ϒ46 ⎥ , ⎥ ϒ46 ⎥ ⎥ ϒ67 ⎦ ϒ77 184 X GUAN ET AL ¯ ϒ33 = (4 − θ)(A + A)T P1 (A + A) − P1 + P2 Proof: Take the following Lyapunov functional: ¯ λ2 I + (τM − τm + 1)Q2 + (6 + θ)ε Vk6 = 12 STk Sk ¯ T LT P1 LC + 3α C T LT P1 LC + 5(1 − θ)C The increment of Vk6 is deduced as follows: ¯ T LT GT (GB)−1 GLC, + 2(1 − θ)C ¯ + ϒ34 = ϒ35 = −θ(A ϒ36 = (A + E{ Vk6 } = E{Vk+1 } − E{Vk6 } A)T P1 LC, = 12 STk+1 Sk+1 − 12 STk Sk ¯ T LT P1 Ad , A)T P1 Ad − (1 − θ)C = 12 STk+1 Sk+1 − 12 STk Sk + STk Sk+1 ¯ T LT P1 LC ϒ44 = 3α C T LT P1 LC + (1 + 6θ)C ¯ L G (GB) + 3θC T T T −1 − STk Sk+1 GLC − P2 , = STk Sk + 12 STk+1 STk − 12 STk STk ¯ L P1 Ad , ϒ46 = −θC T T = STk Sk + ¯ T LT P1 LC + 3α C T LT P1 LC − P2 ϒ55 = 8θC = STk Sk+1 − STk Sk + ¯ T LT GT (GB)−1 GLC, + 2θC with ϒ66 = 2ATd P1 Ad − Q2 , (54) Sk = Sk+1 − Sk Substituting (52) into (9), we have ¯ yk ) − + GL(yck − (1 − θ)ˆ According to Lemmas 2.2–2.3 and letting W = P1 L, it can be shown that the matrix inequality ϒ1 < is ensured by (15) and (16) Hence, if there exist matrices P1 > 0, P2 > 0, Q2 > 0, W > and scalars ε1 > 0, ε2 > satisfying (15) and (16), then the sliding mode dynamic (13) and the error dynamic (14) are stochastically stable = −GAexˆk − sgn(Sk¯ ) Theorem 4.1: Consider the state observer (8) and the sliding surface (9) For the real matrix G ∈ Rm×n and the gain matrix L ∈ Rn×p , assume that the SMC law is synthesized as ¯ yk ) uk = −(GB)−1 [GAˆxk¯ + GL(yck − (1 − θ)ˆ (52) with (55) Therefore, considering the inequality Sk ≤ Sk have E{ Vk6 } = STk Sk + + In this section, the SMC law are designed to attenuate the effects from an event-triggered mechanism and the packet losses such that the system state can arrive at the pre-specified sliding manifold within finite time sgn(Sk¯ )]} ¯ yk ) + GL(yck − (1 − θ)ˆ 2 1, we STk Sk = STk (−GAexˆk − Observer-based sliding mode controller design > GA exˆk + ρ, if STk sgn(Sk¯ ) > 0, ≤ −( GA exˆk + ρ), otherwise, STk Sk , = GAˆxk + GB{−(GB)−1 [GAˆxk¯ ϒ77 = 2ATd P1 Ad − P2 sgn(Sk¯ )], ¯ yk )] Sk+1 = G[Aˆxk + Buk + L(yck − (1 − θ)ˆ ϒ67 = ATd P1 Ad , − STk Sk sgn(Sk¯ )) − STk Sk STk Sk ≤ −ρ Sk − STk Sk + STk Sk (56) Based on the aforementioned discussion, it can be known that Vk6 is negative through regulating the positive parameter ρ to be large enough for Sk = Moreover, Sk is reasonable bounded, which implies that the trajectory of system (1) can be maintained in the pre-defined sliding motion Consequently, the proof is complete Remark 4.1: The features of the main results can be summarized as follows: (1) there is a need to better characterize the packet losses and reflect the induced effects; and (2) the event-triggered mechanism should be addressed properly Accordingly, the Equation (6) is introduced by utilizing the random variable θk , where yck is employed to construct the controller in (52) Moreover, the current ¯ instants are replaced by the trigger instants k (53) where ρ is a positive constant, then the state trajectories can be approached to the sliding manifold within finite time An illustrative example In this section, an illustrative example is presented to show the usefulness of the proposed theoretical results SYSTEMS SCIENCE & CONTROL ENGINEERING: AN OPEN ACCESS JOURNAL 185 Figure State and estimation Consider the system (1) with the following parameter matrices: A= 0.35 0.59 , −0.04 −0.09 Ad = −0.047 −0.68 , 0.002 −0.01 C= 2.5 , H= F = cos(0.3k), B= E= 0.25 , 0.8 0.01 0.01 T , 0.1 0.25 , 0.01 −0.02 f (xk ) = cos(xk1 xk2 ) The initial conditions of original system and state estimator are set as φk = [2 9]T , k ∈ [−5, 0] and xˆ = [0 0]T , respectively Consider the variation range of time delay as τm = and τM = Select the corresponding probability θ¯ = 0.45 Solving the LMIs (15) and (16) by utilizing Matlab LMI toolbox, the observer gain L and other parameters matrices can be obtained as follows: L= P2 = 0.6641 1.8803 T , P1 = 4.7483 5.3530 , 5.3530 88.3364 47.5936 33.5355 , 33.5355 449.9477 δ = 0.41 The simulation results are provided in Figures 2–4 Figure plots the state responses and estimation responses It is observed that the closed-loop system is stochastically stable under the effect of packet losses and event-triggered In addition, the released intervals of the observer to controller are shown in Figure It can be seen Figure Released intervals from the observer to actuator channel that Figure depicts the trajectories of the sliding mode function Sk and the sliding mode controller uk From the simulation, it can be concluded that the proposed SMC law generates desirable control signals to quickly drive the state trajectories onto the sliding surface, which further illustrates the effectiveness of the developed SMC scheme Conclusions In this paper, the SMC problem has been investigated for discrete delayed systems with event-triggered and 186 X GUAN ET AL Figure Control signal uk and trajectories of sliding variable Sk packet losses by designing the observer-based output feedback controller The discrete integral sliding surface has been constructed An SMC law has been synthesized such that the state trajectories of systems are driven onto the neighbourhood of the specified sliding surface Moreover, a sufficient condition has been given to ensure the stochastic stability of the resulting sliding mode dynamics Finally, a simulation example has been given to demonstrate the effectiveness of the proposed control method The further topic motivated by the main results can be listed as: (1) the extension of the method to observer-based SMC for networked systems with different communication transmission protocols as in Shen, Wang, Shen, Alsaadi, and Alsaadi (2020), Liu, Wang, Chen, and Wei (2019), Wang, Wang, Chen, and Sheng (2019), Zou, Wang, Hu, and Gao (2017), Zou, Wang, Hu, and Zhou (2020) and quantized observation in Zou, Wang, Hu, and Han (2020), Liu, Wang, Han, and Jiang (2020); (2) the discussion of the less conservatism of proposed SMC technique handling the delays Disclosure statement No potential conflict of interest was reported by the author(s) Funding This work was supported in part by the National Natural Science Foundation of China [grant number 61673141], the European Regional Development Fund and Sêr Cymru Fellowship [grant number 80761-USW-059], the Outstanding Youth Science Foundation of Heilongjiang Province of China [grant number JC2018001], the Fundamental Research Foundation for Universities of Heilongjiang Province of China and the Alexander von Humboldt Foundation of Germany ORCID Jun Hu http://orcid.org/0000-0002-7852-5064 References Argha, A., Li, L., Su, S., & Nguyen, H (2016) Stabilising the networked control systems involving actuation and measurement consecutive packet losses IET Control Theory and Applications, 10(11), 1269–1280 Chen, S., Guo, J., & Ma, L (2019) Sliding mode observer design for discrete nonlinear time-delay systems with stochastic communication protocol 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(2020) Moving horizon estimation with unknown inputs under dynamic quantization effects IEEE Transactions on Automatic Control doi:10.1109/TAC.2020.2968975 ... Shen, Wang, Shen, Alsaadi, and Alsaadi (2020), Liu, Wang, Chen, and Wei (2019), Wang, Wang, Chen, and Sheng (2019), Zou, Wang, Hu, and Gao (2017), Zou, Wang, Hu, and Zhou (2020) and quantized... sliding 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... Control Theory and Applications, 12(1), 20–28 That, N., & Ha, Q (2015) Discrete- time sliding mode control with state bounding for linear systems with time-varying delay and unmatched disturbances