Passivity-based robust sliding mode synthesis for uncertain delayed stochastic systems via state observer

Automatica 111 (2020) 108596

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Automatica journal homepage: www.elsevier.com/locate/automatica

Brief paper

Passivity-based robust sliding mode synthesis for uncertain delayed stochastic systems via state observer✩ ∗

Zhen Liu a , Hamid Reza Karimi b , Jinpeng Yu a , a b

School of Automation, Qingdao University, Qingdao, 266071, PR China Department of Mechanical Engineering, Politecnico di Milano, Milan, 20156, Italy

article

info

Article history: Received 13 March 2018 Received in revised form 25 July 2019 Accepted 17 August 2019 Available online xxxx Keywords: Sliding mode control Stochastic systems State observer Time-delay

a b s t r a c t In this paper, the problem of passivity-based sliding mode control (SMC) for uncertain delayed stochastic systems (DSS) within state-observer framework is under consideration. A novel linear sliding surface is first presented with the aid of the provided observer and output information, and a new sufficient condition on the passivity and mean square exponential stability of the underlying closedloop system during the sliding mode is then derived by employing the stochastic stability theory and linear matrix inequality approach. Further, an associated SMC law is designed to guarantee the arrival of the sliding surface almost surely via the adaptive methodology. Finally, two illustrative examples concerning the desirable performance of the system are conducted to confirm the effectiveness and superiority of the proposed method. © 2019 Elsevier Ltd. All rights reserved.

1. Introduction The analysis and synthesis problems of uncertain nonlinear systems have attracted increasing attention due to frequent appearances of the uncertainties in physical applications (Liu et al., 2018; Rubagotti, Estrada, Castanos, Ferrara, & Fridman, 2011; Yu, Zhao, Yu, & Lin, 2019; Zhao, Yu, Zhao, Yu, & Lin, 2018). Sliding mode control (SMC), regarded as a typical robust nonlinear control method, has been witnessed as a hot research topic around control community for uncertain/nonlinear systems (Edwards & Spurgeon, 1998). The notable merit of SMC lies in its ability of complete suppression to uncertain variations and external disturbances occurring in the same channel with control signals, owing to the exertion of discontinuous terms in the variable structure controller. Hence, considerable focuses have been devoted to SMC and its applications, see Basin, Ferreira, and Fridman (2007); Li, Shi, and Yao (2017) and the references therein. ✩ This work is supported in part by the National Natural Science Foundation of China (61973179, 61803217, 61573204, 61603204, 61873071), the Natural Science Foundation of Shandong Province, China (ZR2018PF010, ZR2017MF055), the National Key Research and Development Plan, China (2017YFB1303503), the China Postdoctoral Science Foundation (2018M642612), the Shandong Province Outstanding Youth Fund, China (ZR2018JL020), and Taishan Scholar Special Project Fund, China (TSQN20161026). This paper was not presented at any IFAC meeting. This paper was recommended for publication in revised form by Associate Editor Michael V. Basin under the direction of Editor Ian R. Petersen. ∗ Corresponding author. E-mail addresses: [email protected] (Z. Liu), [email protected] (H.R. Karimi), [email protected] (J. Yu). https://doi.org/10.1016/j.automatica.2019.108596 0005-1098/© 2019 Elsevier Ltd. All rights reserved.

SMC has been utilized for uncertain delayed systems in view of the objective situation that time-delays commonly occur in practical control systems, which may lead to unexpected oscillation and system instability (Karimi, 2012; Li & Decarlo, 2003; Liu & Gao, 2016; Wu & Lam, 2008). With respect to both realistic and essential aspects of modeling in practice, these systems are often vitiated by stochastic noises such as Wiener process, the analysis and synthesis of delayed stochastic systems (DSS) have got a growing interest and become more useful for the representation of complex systems (Karimi, Maralani, & Moshiri, 2006; Mao, 2007; Senthilkumar & Balasubramaniam, 2011). During the past years, there have been some researches on SMC for stochastic systems with Wiener process, and some remarkable results can be found in Gao, Feng, Liu, Qiu, and Wang (2014), Huang and Mao (2010), Niu, Ho, and Lam (2005), Niu, Ho, and Wang (2007) and Wu, Gao, Liu, and Li (2017). However, additional restrictions are imposed on system matrix parameters in Niu et al. (2005, 2007), that is, the product of the diffusion term within the stochastic perturbations and the parameter matrices in sliding function must be zero, which will bring more conservatism for such systems. To promote the research scope of stochastic systems, some results are dedicated to remove this strong assumption. The authors in Huang and Mao (2010) firstly proposed an SMC scheme for handling the above problem. It is noteworthy that the controllers and sliding surfaces proposed in the aforementioned reports were under the premise that all the state variables are accessible. Nevertheless, it may be not easy/possible to obtain all of the state components directly for many practical plants. In contrast to the abundant studies on

2

Z. Liu, H.R. Karimi and J. Yu / Automatica 111 (2020) 108596

SMC of DSS by use of completely available information of the state, little progress has been made towards the state estimation problem with SMC for uncertain DSS, which partly motivates us for this study. Moreover, aiming in reducing additional complexity/cost for uncertain systems, the benefits of observer for estimating unmeasured state variables have been well embodied such as Lin, Chen, Shi, and Yu (2018) and Zhao, Yu, and Lin (2019). Consequently, sliding mode observer (SMO) technique (in the sense of observerbased SMC) has been proposed to cope with the state estimation issue for various systems (Aguilar-Lopez & Maya-Yescas, 2005; Li, Shi, Yao, & Wu, 2016; Raoufi, Marquez, & Zinober, 2010; Shi, Liu, & Zhang, 2015; Spurgeon, 2008; Yan, Spurgeon, & Edwards, 2010). On one hand, with regard to the applications of SMO, the architectures for most of the existing works are a kind of non-smooth observers with SMC, which may increase adverse affects caused by the uncertainties, including the case of DSS in Kao, Xie, Wang, and Karimi (2015), Liu and Sun (2012) and Niu and Ho (2006), since observer output variables depend on the discontinuous controllers directly, which may induce high-frequency switching phenomenon on the observer and further influence the precision of the state estimation. On the other hand, the SMO strategy for DSS may not be well displayed yet from the fact that the existing results did not take any perturbations through the same channel with control signals (i.e., matching uncertainty in SMC theory) into consideration. The technical obstacles can be summarized as: (1) How to give a suitably smooth observer scheme such that the estimation of the unmeasured state variables will be properly guaranteed without certain negative impacts resulting from discontinuous terms of the previous observers; (2) How to synthesize an associated adaptive controller to suppress the unknown matching perturbations and potential uncertainties by incorporating the proposed SMO. Therefore, an investigation into SMO of uncertain DSS is of great significance and remains open and challenging, which also motivates our present study. Furthermore, passivity analysis has received extensive attention in control theory and applications (Liang, Wang, & Liu, 2010; Wu & Zheng, 2009), which is often defined in view of energy dissipation and its transformation. As a fundamental performance, many control systems may be passive so as to attenuate potential environmental disturbances. To the best of the authors’ knowledge, related reports on passivity-based control of DSS via SMO have not been shown yet. Inspired by the above representations, the passivity-based stabilization problem for uncertain DSS subject to matching perturbations, modeling uncertainties and external disturbances is studied by employing a novel SMO framework in this paper. In comparison to the relevant works (Gao et al., 2014; Huang & Mao, 2010; Kao et al., 2015; Liu & Sun, 2012; Niu & Ho, 2006; Niu et al., 2005, 2007), the main contributions of the paper are listed below: (1) A state observer with a simplified architecture is presented without any non-smooth control terms for the DSS. The discontinuous terms existing in the traditional SMOs are no longer required and only one parameter is needed; (2) By virtue of the observer and output information, a novel sliding surface of linear type is proposed, and a new sufficient condition for the resultant closed-loop system to be exponentially stable in mean square sense is derived with the help of special structure of the designed sliding surface; (3) The unknown matching perturbations, modeling uncertainties and disturbance inputs can be well steered by the developed new adaptive sliding mode controller, guaranteeing the expected performance of the closed-loop system. Also, the restrictive assumptions related to sliding function and correlated matrix parameters as well as the diffusion function terms in Niu and Ho (2006), Niu et al. (2005, 2007) and Liu and Sun (2012) are resolved in this study.

The rest of this paper is organized as follows. Section 2 presents system description. Section 3 contains the main result of this paper by providing the observer-based SMC scheme. Section 4 shows the simulation studies and Section 5 ends the paper with some conclusions. Notation: For real symmetric matrices X and Y , X > Y represents that X − Y is a positive definite matrix. (Ω , F , {Ft }t ≥0 , P ) denotes a completed probability space with a natural filtration {Ft }t ≥0 , Ω is a sample space, F is the σ -algebra of the space subset, and P is the probability measure defined on F . E {·} stands for the expectation operator concerning the probability measure P . In is used to represent the n × n identity matrix. ‘T’ is denoted as the transpose of a matrix or vector, and ‘‘∗’’ stands for the symmetric elements of a matrix. sym{Y } is shown as sym{Y } = Y + Y T . Tr(·) is denoted as the trace of a matrix. λmax (P) represents the maximum eigenvalue of matrix P. diag {·} stands for a block-diagonal matrix. ∥ · ∥ denotes∑the Euclidean n vector norm or spectral matrix norm, and ∥x∥1 ≜ i=1 |xi | stands n for the 1-norm of a vector x, where x = (xi ) ∈ R . 2. System statement and preliminary Consider a class of delayed stochastic systems (DSS) given by the following equations:

⎧ ⎪ ⎪ dx(t) = {(A + ∆A(t))x(t) + (Aτ + ∆Aτ (t))x(t − τ (t)) ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ +B[u(t) + g(t , x)] + Dw w(t)}dt + h(t , x)dω(t), ⎨ y(t) = Cx(t), ⎪ ⎪ ⎪ ⎪ ⎪z(t) = Ex(t) + F w(t), ⎪ ⎪ ⎪ ⎩ x(θ ) = φ (θ ), θ ∈ [−τ , 0],

(1)

where the state x(t) ∈ Rn , control input u(t) ∈ Rm , measured output y(t) ∈ Rp and controlled output z(t) ∈ Rl . ω(t) is a standard scalar Wiener process specified on (Ω , F , {Ft }t ≥0 , P ) with a natural filtration {Ft }t ≥0 and meets E {dω(t)} = 0 and E {dω2 (t)} = dt. τ > 0 is the upper bound of time-varying state delay τ (t) satisfying τ˙ (t) ≤ β < 1. A, Aτ , B, C , Dw , E and F are constant matrices with B of full column rank. w (t) ∈ Rq is the exogenous input belonging to L2 [0, ∞), which is an unknown function and bounded as ∥w (t)∥ ≤ ϑ , where ϑ > 0 is an unknown scalar. The parameter uncertainties ∆A(t) and ∆Aτ (t) are norm-bounded, i.e., [∆A(t) ∆Aτ (t)] = MJ(t)[N Nτ ], where J(t) is an unknown function which satisfies J T (t)J(t) ≤ I, M, N and Nτ are real constant matrices. φ (t) is the initial function. The matching uncertain function g(t , x) is unknown and satisfies ∥g(t , x)∥ ≤ a∥y(t)∥ + b, where a and b are unknown positive scalars. h(t , x) denotes the diffusion gain function, which is not exactly known and bounded by the output signal. As seen from Mao (2007), it is justified that the existence and uniqueness of the solution in system (1) are ensured with u(t) = 0 and w (t) = 0. Definition 1 (Mao, 2007). If a positive scalar λ exists to guar2 antee limt →∞ sup 1 t logE {∥x(t , φ )∥ } ≤ −λ for all admissible uncertainties, then the system in (1) is said to be mean square exponentially stable with u(t) = 0 and w (t) = 0. Definition 2 (Liang et al., 2010; Wu & Zheng, 2009). The system in (1) is said to be passive in the sense of expectation if a scalar γ > 0 exists to satisfy 2E

{∫

}

t∗

z (t)w (t)dt T

0

≥ −γ E

{∫

}

t∗

w (t)w(t)dt T

(2)

0

for all t ∗ > 0 and admissible uncertainties under zero initial conditions.

Z. Liu, H.R. Karimi and J. Yu / Automatica 111 (2020) 108596

3

Lemma 1 (Huang & Mao, 2010). Given two real matrices Q ∈ Rp×p and R ∈ Rp×q with Q ≥ 0, it follows that

3.2. Stability analysis of closed-loop system with passivity

Tr(RT QR) ≤ λmax (Q )Tr(RT R).

This part is focused on the desirable performance (i.e., exponential stability with passivity in mean square sense) of the closed-loop system once all the system trajectories arrive at the designed sliding surface almost surely.

3. Observer-based SMC design This section aims at solving the observer-based SMC problem for uncertain DSS in (1) with passivity performance subject to matching perturbations, structural uncertainties and disturbance inputs. Specifically, the following research topics are addressed in this section: (a) Novel smooth-type state observer design; (b) A new sliding surface establishment of linear type; (c) Performance analysis of the resultant closed-loop system; (d) An associated novel adaptive switching controller synthesis guaranteeing the existence ability of (b) and desirable performance of (c) under investigation. 3.1. Observer and linear-type sliding surface design In view of the discontinuous switching controller u(t) utilized in most of the existing reports, we first give a modified state observer for generating the unknown components of the system state so as to avoid direct impact of high-frequency switching operations on the observer performance. The non-fragile observer of the plant in (1) is given with the state estimation xˆ (t) ∈ Rn , observer output yˆ (t) ∈ Rp as follows:

⎧ dxˆ (t) = {Axˆ (t) + Aτ xˆ (t − τ (t)) ⎪ ⎪ ⎪ ⎨ +(L + ∆L(t))[y(t) − C xˆ (t)]}dt , yˆ (t) = C xˆ (t), ⎪ ⎪ ⎪ ⎩ xˆ (θ ) = φˆ (θ ), θ ∈ [−τ , 0],

(3)

L ∈ R is the designed gain matrix of the observer, and ∆L(t) may represent the additional gain variation satisfying ∥∆L(t)∥ ≤ q with q > 0, i.e., the observer may be perturbed by unknown conditions. φˆ (θ ) is the initial function of the observer. Herein, we denote the state error variable by e(t) = x(t) − xˆ (t). Thus, from (1) and (3), it follows the estimation error system: n×p

where matrix Y ∈ R

m×p

dx(t) = {[A + BK + ∆A(t)]x(t) + [Aτ + ∆Aτ (t)]x(t − τ (t))

ˆ , x)] + Dw w(t)}dt + h(t , x)dω(t), + B[u(t) + g(t de(t) = {[BK + ∆A(t)]x(t) + ∆Aτ (t)x(t − τ (t)) + [A − LC ˆ , x)] − ∆L(t)C ]e(t) + Aτ e(t − τ (t)) + B[u(t) + g(t + Dw w(t)}dt + h(t , x)dω(t),

(6)

(7)

ˆ , x) = g(t , x) − Kx(t), and K is a parametric matrix where g(t to meet that A + BK is Hurwitz. Besides, it is assumed that a parametric matrix H may exist to meet that Tr{hT (t , x)h(t , x)} ≤ ∥Hy(t)∥2 holds. [ ] x(t) Denoting ξ (t) = , the augmented closed-loop system e(t) with (6) and (7) can be described by ˆ , x)dω(t), + h(t

(4)

Herein, the following novel sliding surface function of lineartype is given by s(t) = Y [2y(t) − C xˆ (t)],

In this position, the overall systems consisting of the original system in (1), the observer in (3) and the error system in (4) are recalled by the following representations:

ˆ ξ (t) + Aˆ τ (t)ξ (t − τ (t)) + Bˆ w (t , x)]dt dξ (t) = [A(t)

de(t) = {∆A(t)x(t) + ∆Aτ (t)x(t − τ (t)) + [A − LC

− ∆L(t)C ]e(t) + Aτ e(t − τ (t)) + B[u(t) + g(t , x)] + Dw w (t)}dt + h(t , x)dω(t).

Remark 3. As is known, if the observer and the state error tend to be stable, then stability of the overall system will be guaranteed. This idea has been widely used in stability analysis by the observer-based method, including the observer-based SMC for DSS. From another perspective, if we can ensure the performance of the system and the error part, the observer will be stable also. In this paper, we will use this new analysis path to expand and further explore the performance of the closed-loop system based on the proposed SMO, which makes the addressed method differ from the current works, see Kao et al. (2015), Li et al. (2016), Liu and Sun (2012), Niu and Ho (2006), Raoufi et al. (2010) and Yan et al. (2010).

(5)

is to be determined.

Remark 1. In this subsection, a simplified continuously observer is presented to generate the unmeasured state variables in (1). It is observed that discontinuous terms existing in the traditional SMOs are no longer required and only one parameter is needed, i.e., observer gain matrix L here, thus distinguishing it from the current studies (Kao et al., 2015; Liu & Sun, 2012; Niu & Ho, 2006). It is known that undesirable phenomenon called ‘chattering’ in variable structural systems may frequently be exacerbated due to the non-smooth control terms in previous SMOs, while the key shortcoming of the SMO system can be avoided via the provided scheme in some degree. Remark 2. Compared with the existing works on SMC-based strategy for DSS (Liu & Sun, 2012; Niu & Ho, 2006), the linear sliding surface in (5) is much easier to design, which depends on the current output signals only. Moreover, this design from system analysis prospective, could facilitate the following deduction of stability analysis of the closed-loop system combining condition (11) in the following Theorem 1.

(8)

where

[ ] 0 ˆA(t) = A + BK + ∆A(t) , BK + ∆A(t) A − LC − ∆L(t)C [ ] [ ] ˆAτ (t) = Aτ + ∆Aτ (t) 0 , h(t ˆ , x) = h(t , x) , ∆Aτ (t) Aτ h(t , x) [ ] ˆ , x)) + Dw w(t) B(u(t) + g(t Bˆ w (t , x) = . ˆ , x)) + Dw w(t) B(u(t) + g(t Remark 4. It should be noted that the resultant closed-loop system is rewritten concerning the introduced parametric matrix K . It is worth pointing out that K is not the so-called feedback gain, because there is no connection between K with the controller u(t). In detail, the purpose of introducing the matrix K is only correlated to the stability analysis of the associated sliding motion so as to achieve the feasibility of the determined parameters. Theorem 1. Given a scalar γ > 0, the exponential stability of the closed-loop system restricted on the linear sliding surface s(t) = 0 with passivity performance is guaranteed in mean square sense, if conditions (9)–(11) are satisfied with positive definite matrices X , U1 , U2 , matrices Y and Z , scalars κ > 0 and λi > 0, i = 1, . . . , 3 as follows X < κI,

(9)

4

Z. Liu, H.R. Karimi and J. Yu / Automatica 111 (2020) 108596

⎡ Υ11 ⎢ ∗ ⎢ ⎢ ∗ ⎢ ⎢ ⎢ ∗ ⎢ ⎣ ∗ ∗

Υ12 Υ22 ∗ ∗

∗ ∗

(XBK )T

0

XDw − E T

0

0

0

Υ33 ∗

XAτ

XDw

Υ44

0

∗ ∗

∗ ∗

Π1

and

⎤

−2eT (t)X ∆L(t)Ce(t) 1 T 2 T T ≤ λ− 3 e (t)XXe(t) + λ3 q e (t)C Ce(t).

0⎥ ⎥

−γ I − F − F ∗

T

Π2 ⎥ ⎥ ⎥ < 0, 0⎥ ⎥ 0⎦

(10)

Π3

BT X = YC ,

(11)

where Υ11 = sym{X (A + BK )} + U1 + (λ1 + λ2 )N T N + 2κ (HC )T HC , Υ12 = XAτ + (λ1 + λ2 )N T Nτ , Υ22 = −(1 − β )U1 + (λ1 + λ2 )NτT Nτ , Υ33 = sym{XA − ZC } + U2 + λ3 q2 C T C , Υ44 = −(1 − β )U2 , Π1 = [XM 0 0], Π2 = [0 XM X ], Π3 = diag {−λ1 I , −λ2 I , −λ3 I }. Moreover, the gain matrix of the observer is given as L = X −1 Z . Proof. Firstly, the passivity of the closed-loop system is considered. Choose a Lyapunov–Krasovskii functional as V (ξ (t), t) = ξ T (t)Xˆ ξ (t) +

= xT (t)Xx(t) +

t

∫

ξ T (θ )Uˆ ξ (θ )dθ t −τ (t) t

∫

+ eT (t)Xe(t) +

Since the underlying system in (8) is attracted and maintained on the designed sliding surface, namely, s(t) = Y [2y(t) − C xˆ (t)] = BT X [x(t) + e(t)] = 0 according to (11), it also follows sT (t) = [xT (t) + eT (t)]XB = 0. Then, by adding (12)–(14), one can get LV (x(t), e(t), t) 1 T ≤ xT (t)[sym{X (A + BK )} + U1 + λ− 1 XMM X ]x(t)

+2xT (t)XAτ x(t − τ (t)) + 2xT (t)XDw w(t) +2Tr{hT (t , x)Xh(t , x)} + eT (t)[sym{X (A − LC )} 1 −1 T 2 T +U2 + λ− 2 XMM X + λ3 XX + λ3 q C C ]e(t) +2eT (t)XAτ e(t − τ (t)) + 2eT (t)XBKx(t) +2eT (t)XDw w(t) + λ1 [Nx(t) + Nτ x(t − τ (t))]T ·[Nx(t) + Nτ x(t − τ (t))] + λ2 [Nx(t) +Nτ x(t − τ (t))]T [Nx(t) + Nτ x(t − τ (t))] −(1 − β )xT (t − τ (t))U1 x(t − τ (t)) −(1 − β )eT (t − τ (t))U2 e(t − τ (t)). Tr{hT (t , x)Xh(t , x)} ≤ λmax (X )∥Hy(t)∥2

≤ κ xT (t)(HC )T HCx(t).

eT (θ )U2 e(θ )dθ t −τ (t)

where Xˆ = diag {X , X }, Uˆ = diag {U1 , U2 }. According to the Itoˆ formula, the stochastic differential of V (ξ (t), t) becomes

LV (x(t), e(t), t) − 2z T (t)w (t) − γ w T (t)w (t)

≤ ϕ T (t)Ω ϕ (t),

(17)

where ϕ (t) = [x (t) x (t − τ (t)) e (t) e (t − τ (t)) w (t)], T

ˆ , x)dω(t) dV (ξ (t), t) = LV (ξ (t), t) + 2ξ T (t)Xˆ h(t

T

Ω11 ⎢ ∗ ⎢ ⎢ Ω=⎢ ∗ ⎢ ⎣ ∗

⎡

= LV (ξ (t), t) + 2[x(t) + e(t)]T Xh(t , x)dω(t) where

ˆ ξ (t) + Aˆ τ (t)ξ (t − τ (t)) LV (ξ (t), t) = 2ξ T (t)Xˆ [A(t) ˆ , x)} + Bˆ w (t , x)] + Tr{hˆ T (t , x)Xˆ h(t T + ξ (t)Uˆ ξ (t) − (1 − τ˙ (t))ξ T (t − τ (t)) ·Uˆ ξ (t − τ (t)) = 2xT (t)X {[A + BK + ∆A(t)]x(t) + [Aτ + ∆Aτ (t)]x(t − τ (t)) + Dw w(t)} + xT (t)U1 x(t) − (1 − τ˙ (t))xT (t − τ (t)) ·U1 x(t − τ (t)) + 2Tr{hT (t , x)Xh(t , x)} + 2eT (t)X {[A − LC − ∆L(t)C ]e(t) + Aτ e(t − τ (t)) + [BK + ∆A(t)]x(t) +∆Aτ x(t − τ (t)) + Dw w(t)} + eT (t)U2 e(t) −(1 − τ˙ (t))eT (t − τ (t))U2 e(t − τ (t)) ˆ , x)] +2[xT (t) + eT (t)]XB[u(t) + g(t = LV (x(t), e(t), t).

T

T

T

T

Ω12 Ω22 ∗ ∗

(XBK )T

0

XDw − E T

⎤

0

0

0

Ω33 ∗

XAτ

XDw

Ω44

0

⎥ ⎥ ⎥ ⎥, ⎥ ⎦

∗

∗

∗

−γ I − F − F T

∗

1 T T where Ω11 = sym{X (A + BK )}+ U1 +λ− 1 XMM X + (λ1 +λ2 )N N +

2κ (HC )T HC , Ω12 = XAτ + (λ1 + λ2 )N T Nτ , Ω22 = −(1 − β )U1 + −1 1 T (λ1 + λ2 )NτT Nτ , Ω33 = sym{XA − ZC } + U2 + λ− 2 XMM X + λ3 XX + 2 T λ3 q C C with Z = XL, and Ω44 = −(1 − β )U2 . By taking condition (10) and the Schur complement into consideration, it follows that LV (x(t), e(t), t) − 2z T (t)w (t) − γ w T (t)w (t) ≤ 0.

(18)

Moreover, by the Dynkin’s formula, one has E {V (x(t), e(t), t)} = E

t

{∫

LV (x(θ ), e(θ ), θ )dθ

} (19)

0

under zero initial conditions. Then, it follows that E

{∫

}

t∗

[−2z (θ )w (θ ) − γ w (θ )w(θ )]dθ T

T

0

In addition, the following inequalities hold 2x (t)X [∆A(t)x(t) + ∆Aτ (t)x(t − τ (t))]

≤E

T

{∫

T

·[Nx(t) + Nτ x(t − τ (t))],

}

t∗

LV (x(θ ), e(θ ), θ )dθ 0

≤ λ1 x (t)XMM Xx(t) + λ1 [Nx(t) + Nτ x(t − τ (t))] T

(16)

By substituting (16) into (15), it follows that

= V (x(t), e(t), t),

−1 T

(15)

In view of condition (9) and Lemma 1, it follows that

xT (θ )U1 x(θ )dθ t −τ (t) t

∫

(14)

(12)

−E

{∫

}

t∗

[2z (θ )w(θ ) + γ w (θ )w(θ )]dθ T

T

0

2eT (t)X [∆A(t)x(t) + ∆Aτ (t)x(t − τ (t))]

≤ 0,

1 T T T ≤ λ− 2 e (t)XMM Xe(t) + λ2 [Nx(t) + Nτ x(t − τ (t))]

·[Nx(t) + Nτ x(t − τ (t))],

(13)

which ensures the passivity of the system for all t Definition 2.

(20) ∗

> 0 from

Z. Liu, H.R. Karimi and J. Yu / Automatica 111 (2020) 108596

Next, stability analysis of the discussed system is investigated under w (t) = 0. Obviously, the inequality in (17) is changed into the following expression:

˜ ψ (t), LV (x(t), e(t), t) ≤ ψ T (t)Ω

(21)

where ψ T (t) = [xT (t) xT (t − τ (t)) eT (t) eT (t − τ (t))], and

⎡ Ω11 ⎢ ∗ ˜ Ω=⎣ ∗ ∗

Ω12 Ω22 ∗ ∗

(XBK )T 0

Ω33 ∗

Remark 5. In Theorem 1, a new sufficient exponential stability condition with passivity for the sliding motion in mean square sense is established by resorting to the arrival of the sliding surface s(t) = 0. Moreover, it is noted that the conditions in (9)–(10) and the equality condition in (11) are interconnected, which may bring difficulty in solving the parameters X , Y , Z , etc. To reduce the computational complexity and obtain the feasible solutions with Matlab software, the following optimization algorithm is introduced: Step 1: Transform the condition BT X = YC into the following equality Tr{(BT X − YC )T (BT X − YC )} = 0,

(22)

it follows that (BT X − YC )T (BT X − YC ) < µI

(23)

for any sufficiently small positive scalar µ. Step 2: Using the Schur complement, (23) is equivalently changed into

]

< 0.

(24)

To this end, one can get a possible choice for replacing the combined conditions by the following approximation problem (Niu et al., 2005): min µ, s.t. (9) (10) and (24). The equality condition and feasibility of Theorem 1 will be guaranteed in synchrony once the globally feasible solution µ equal to or is sufficiently small to approach zero. 3.3. Finite-time reachability of sliding surface In this subsection, the emphasis lies in designing an appropriate controller such that the system trajectories will be attracted onto the predesigned linear sliding surface almost surely. Notice that the solutions of systems (1) and (4) are, respectively, represented by t

∫

f1 (θ, x)dθ +

x(t) = x(0) +

t

∫

0

h(θ, x)dω(θ ),

(25)

0

t

f2 (θ, x)dθ +

e(t) = e(0) + 0

t

∫

h(θ, x)dω(θ ), 0

where Ls(t) = B X {Ax(t) + 2∆A(t)x(t) + 2Aτ x(t − τ (t)) − Aτ xˆ (t − τ (t)) + 2∆Aτ (t)x(t − τ (t)) + [A − LC − ∆L(t)C ]e(t) + 2B[u(t) + g(t , x)] + 2Dw w (t)}. In practical cases, one may not catch the exact representation of unknown perturbations in the same channel with the controller, the adaptive control algorithm will be applied to track the associated upper bounds so that the involved uncertainties can be inhibited. Further, without loss of generality, assume that the system is detectable and stabilizable by default, which means the output should be bounded and finite for bounded and finite input at each time via certain proper control scheme, combining the linear dependence of system state x(t) and measured output signal y(t), the following estimation will be given as ∥x(t)∥ ≤ l1 ∥y(t)∥ + l2 , where li > 0, i = 1, 2, are unknown. Assumption 1 (Li & Decarlo, 2003; Wu & Lam, 2008). An unknown scalar l3 > 0 with the following inequality may be found in view of the finite time-delay that

∥x(t − τ (t))∥ ≤ l3 ∥x(t)∥.

(26)

(28)

Then, from the above statement, it is reasonably shown that unknown positive scalars ρi , i = 1, 2, exist to satisfy the following inequalities:

χ (t) = 2∥BT XA∥∥x(t)∥ + 2∥BT X ∆A(t)∥∥x(t)∥ + 2∥BT XAτ ∥∥x(t − τ (t))∥ + 2∥BT X ∆Aτ (t)∥ ·∥x(t − τ (t))∥ + 2∥BT XB∥∥g(t , x)∥ + 2∥BT XDw ∥w(t) ≤ 2∥BT XA∥(l1 ∥y(t)∥ + l2 ) + 2∥BT XM ∥∥N ∥ ·(l1 ∥y(t)∥ + l2 ) + 2l3 ∥BT XAτ ∥(l1 ∥y(t)∥ + l2 ) + 2l3 ∥BT XM ∥∥Nτ ∥(l1 ∥y(t)∥ + l2 ) + 2∥BT XB∥ ·(a∥y(t)∥ + b) + 2∥BT XDw ∥ϑ ≤ ρ1 ∥y(t)∥ + ρ2 , t ≥ 0. (29) Theorem 2. Consider that the linear sliding function is proposed by (5) and matrix L is solved by Theorem 1, the arrival of the designed sliding surface is guaranteed in finite-time almost surely, if the following adaptive controller is utilized:

⎧ 0.5(BT XB)−1 BT X [Axˆ (t) + L(y(t) − C xˆ (t)) + Aτ ⎪ ⎪ ⎪ ⎪ ⎪ ·ˆx(t − τ (t))] − 0.5(BT XB)−1 [q∥BT X ∥∥(y(t) − C xˆ (t))∥ ⎪ ⎪ ⎪ ⎪ ⎨ ∥Hy(t)∥2 +ρˆ 1 (t)∥y(t)∥ + ρˆ 2 (t) + 4λmax ((BT X )T BT X ) ∥s(t)∥ u(t) = ⎪ ⎪ +σ ]sgn(s(t)), if s(t) ̸ = 0; ⎪ ⎪ ⎪ ⎪ T −1 T ⎪ 0.5(B XB) B X [Axˆ (t) + L(y(t) − C xˆ (t)) ⎪ ⎪ ⎩ +Aτ xˆ (t − τ (t))], if s(t) = 0, where ρˆ i (t), i = 1, 2, denote the tracking parameters of ρi , and the error terms are represented by ρ˜ i (t), and the updating laws are given by

{ υ1 ∥y(t)∥, ρ˙ˆ 1 (t) = 0,

if s(t) ̸ = 0; if s(t) = 0,

and

ρ˙ˆ 2 (t) =

and

∫

(27)

T

Ω44

(BT X − YC )T −I

where f1 (θ, x) = (A + ∆A(θ ))x(θ ) + (Aτ + ∆Aτ (θ ))x(θ − τ (θ )) + B[u(θ ) + g(θ, x)] + Dw w (θ ), f2 (θ, x) = (A − LC − ∆L(θ )C )e(θ ) + Aτ e(θ − τ (θ )) + ∆A(θ )x(θ ) + ∆Aτ (θ )x(θ − τ (θ )) + B[u(θ ) + g(θ, x)] + Dw w (θ ), and the last terms of (25) and (26) are the Itoˆ stochastic integrals. Thus, the sliding function is still an Itoˆ stochastic integral with the stochastic differential given by ds(t) = BT X d[x(t) + e(t)] = Ls(t)dt + 2BT Xh(t , x)dω(t),

⎤

0 0 ⎥ ⎦. XAτ

˜ < 0 by It is observed that the condition in (10) leads to Ω the Schur complement. Introduce an auxiliary function F (t) = eϱt V (ξ (t), t) with the infinitesimal operator L given as LF (t) = ϱeϱt V (ξ (t), t) + eϱt LV (ξ (t), t). To this end, it is verified that the dynamics of the sliding motion are mean square exponentially stable from Definition 1 in view of the line similar to that in Huang and Mao (2010) and Chap. 4 of Mao (2007).

[ −µI ∗

5

{

υ2 , 0,

if s(t) ̸ = 0; if s(t) = 0,

with ρˆ 1 (0) = 0, ρˆ 2 (0) = 0; and υi > 0, i = 1, 2, and σ > 0 are constants.

6

Z. Liu, H.R. Karimi and J. Yu / Automatica 111 (2020) 108596

+ υ2−1 ρ˜ 2 (t)ρ˙˜ 2 (t)

Proof. The following function is considered for all t ≥ 0 V (t , s(t)) = (sT (t)s(t))0.5 + 0.5[υ1−1 ρ˜ 12 (t) + υ2−1 ρ˜ 22 (t)]

= −σ ,

= ∥s(t)∥ + 0.5[υ1−1 ρ˜ 12 (t) + υ2−1 ρ˜ 22 (t)]. sT (t)BT Xh(t , x)

∥s(t)∥

E ∥s(t)∥ ≤ E V (t , s(t)) ≤ E V (0, s(0)) − σ t ,

dω(t),

thus, it follows ∥s(t)∥ = 0 almost surely for all t ≥ tf = thereby completing the proof.

where LV (t , s(t)) =

sT (t)

BT X {2Ax(t) + 2∆A(t)x(t) + 2Aτ x(t − τ (t)) ∥s(t)∥ − Aτ xˆ (t − τ (t)) + 2∆Aτ (t)x(t − τ (t)) − Axˆ (t) − L(y(t) − C xˆ (t)) − ∆L(t)(y(t) − C xˆ (t)) + 2B[u(t) + g(t , x)] + 2Dw w(t)} ( ) Im s(t)sT (t) + 2hT (t , x)(BT X )T − ∥s(t)∥ ∥s(t)∥3 ·BT Xh(t , x) + υ −1 ρ˜ 1 (t)ρ˙˜ 1 (t) 1

+ υ2−1 ρ˜ 2 (t)ρ˙˜ 2 (t),

if s(t) ̸ = 0.

(30)

In the light of the designed adaptive controller and considering ∥s(t)∥ ≤ ∥s(t)∥1 , it yields LV (t , s(t)) ≤

≤

sT (t)

B X {2Ax(t) + 2∆A(t)x(t) + 2Aτ x(t − τ (t)) ∥s(t)∥ + 2∆Aτ (t)x(t − τ (t)) − ∆L(t)(y(t) − C xˆ (t)) + 2Dw w(t) + 2Bg(t , x) − B(BT XB)−1 ·[q∥BT X ∥∥(y(t) − C xˆ (t))∥ + ρˆ 1 (t)∥y(t)∥ + ρˆ 2 (t) ∥Hy(t)∥2 + 4λmax ((BT X )T BT X ) + σ ]sgn(s(t))} ∥s(t)∥ hT (t , x)(BT X )T BT Xh(t , x) +4 + υ1−1 ρ˜ 1 (t)ρ˙˜ 1 (t) ∥s(t)∥ + υ2−1 ρ˜ 2 (t)ρ˙˜ 2 (t) T

1

∥s(t)∥{2∥BT XA∥∥x(t)∥ + 2∥BT X ∆A(t)∥ ∥s(t)∥ ·∥x(t)∥ + 2∥BT XAτ ∥∥x(t − τ (t))∥ + 2∥BT X ∆Aτ (t)∥ ·∥x(t − τ (t))∥ + 2∥BT XB∥∥g(t , x)∥ + 2∥BT XDw ∥ϑ +∥BT X ∆L(t)(y(t) − C xˆ (t))∥} −

sT (t)

∥s(t)∥ ·[q∥BT X ∥∥(y(t) − C xˆ (t))∥ + ρˆ 1 (t)∥y(t)∥ + ρˆ 2 (t) ∥Hy(t)∥2 + 4λmax ((BT X )T BT X ) + σ ]sgn(s(t)) ∥s(t)∥ hT (t , x)(BT X )T BT Xh(t , x) + υ1−1 ρ˜ 1 (t)ρ˙˜ 1 (t) +4 ∥s(t)∥ + υ2−1 ρ˜ 2 (t)ρ˙˜ 2 (t) ≤ 2∥BT XA∥∥x(t)∥ + 2∥BT X ∆A(t)∥∥x(t)∥ + 2∥BT XAτ ∥∥x(t − τ (t))∥ + 2∥BT X ∆Aτ (t)∥ ·∥x(t − τ (t))∥ + 2∥BT XB∥∥g(t , x)∥ + 2∥BT XDw ∥ϑ − [ρˆ 1 (t)∥y(t)∥ + ρˆ 2 (t) ∥Hy(t)∥2 + 4λmax ((BT X )T BT X ) + σ] ∥s(t)∥ hT (t , x)(BT X )T BT Xh(t , x) +4 ∥s(t)∥ −1 ˙ + υ1 ρ˜ 1 (t)ρ˜ 1 (t) + υ2−1 ρ˜ 2 (t)ρ˙˜ 2 (t), if s(t) ̸ = 0. In addition, it follows that ρ˙˜ i (t) = ρ˙ˆ i (t), i = 1, 2. By virtue of Lemma 1, one further has LV (t , s(t)) ≤ −ρ˜ 1 (t)∥y(t)∥ − ρ˜ 2 (t) − σ + υ1 ρ˜ 1 (t)ρ˙˜ 1 (t) −1

(31)

To this end, by implementing the time integrals on both sides of (31) and the mathematical expectations, it yields

From the Itoˆ formula, it follows that dV (t , s(t)) = LV (t , s(t))dt + 2

if s(t) ̸ = 0.

E V (0, s(0))

σ

,

Remark 6. It is worthy of note that the developed controller can not only accommodate the attraction of the system trajectories onto the designed sliding surface but also handle the unknown uncertainties perturbed in the system via the proposed adaptive terms ρˆ 1 (t) and ρˆ 2 (t) without considering the prior exact knowledge of the unknown bounds ρ1 and ρ2 . By means of the pertinent Lyapunov function, strong restrictions on system matrix parameters (i.e., product of the diffusion term within the stochastic perturbation and the appropriate parameter matrix in the sliding function of the system must be zero) among the existing reports on SMC of DSS no longer need to be investigated despite the unmeasured state variables. Remark 7. It is noted that the works in Liu and Sun (2012) and Niu and Ho (2006) did not consider any perturbations in the same channel with control signals due to some difficulty. As mentioned, the methods in Gao et al. (2014), Huang and Mao (2010) and Niu et al. (2005, 2007) cannot be workable for more general cases of the DSS subject to unmeasured state variables, which also indicates the novelty of the presented scheme in this paper. Subsequently, the effectiveness of the developed method is verified via two simulation examples. 4. Illustrative examples Example 1 (Gao et al., 2014; Huang & Mao, 2010). The waterquality dynamic system model encountered by environmental stochastic noise and external disturbance is introduced as the description expressed in (1). The following is devoted to using the proposed control scheme to the DSS model. The system state x(t) = [xT1 (t) xT2 (t)]T , where x1 (t) and x2 (t) are denoted as algae and ammonia products, respectively, which are the concentrations of two main sorts of pollutant sources; u(t) stands for the action of the applied controller. Assume that the concentrations of partial or all the water quality components cannot be exactly measured in real time. Simulation data are borrowed from Gao et al. (2014) with some modifications as follows:

[ ] [ ] [ ] −1 1 0 −0.05 1 A= , Aτ = ,B = , −2 −3 0.25 0.5 0 [ ]T 0.1 C = , −0.1 [ ] [ ] 0.25 0 0 0 0 0 0 0.4 M= , N = Nτ = , 0 0 0 0 0 0.2 0.2 0.2 [ ] 0.5sin(2t) 0 0 0 0 0 J(t) = . 0 0 0 0 0 0.4sin(3t) The matching uncertain function g(t , x), diffusion function h(t , x), observer perturbation ∆L(t) and external disturbance w (t) are chosen as

[ √

g(t , x) = − 3cos(t) h(t , x) =

[

0.5 −0.1

0.6cos2 (x1 (t)) + 1 x(t),

]

] [ ] −0.5 0.1sin(t) x(t), ∆L(t) = , 0.1 0

and w (t) = −2cos(2t)e−0.5t .

Z. Liu, H.R. Karimi and J. Yu / Automatica 111 (2020) 108596

Fig. 1. Time response of the system state.

7

Fig. 2. Time response of the state estimation.

The parametric matrix K is chosen as K = −2.2225 −1.2215 . By solving the optimization problem in Remark 5, the feasible solutions are obtained for γ = 1.2126, τ = 0.5, q = 0.1, [ ]T [ ] [ ]T Dw = −0.1 0 , E = −0.4 0 , F = 0.4, H = 5 −1 as follows:

[

]

82.3240 −82.3240

] [ ] −82.3240 22.2543 25.2711 , U1 = , 165.9744 25.2711 295.1361 [ ] [ ] 446.8931 −194.4719 1.7548 U2 = , Z = 104 · , −194.4719 453.5958 −1.1217 [

X =

Y = 823.2398, κ = 220.4682, λ1 = 182.5227,

λ2 = 140.0152, λ3 = 658.6679 with µ = 5.1247 · 10−8 , and the equality condition in (11) can be satisfied. [ Then, the gain matrix L of the observer is obtained as L = 288.8381

Fig. 3. Curve of the linear sliding function.

]T

75.6819 . In the simulation experiment, the initial conditions are given as x(θ ) = [0.5 − 0.5]T , and xˆ (θ ) = [0.5 0]T , θ ∈ [−0.5, 0]. By utilizing the designed observer in (3), linear sliding function in (5) and adaptive controller u(t) with σ = 1.2, υ1 = 2.5 and υ2 = 0.5, simulation results along 11 individual Wiener process paths are displayed by Figs. 1–5. It can be seen that the system can be operated stably despite the unmeasurable state variables, matching perturbations and potential uncertainties, which implies the availability and superiority of the proposed observer-based SMC method in comparison to Gao et al. (2014), Huang and Mao (2010), Niu and Ho (2006), and Niu et al. (2005, 2007). Example 2. Consider a three-order uncertain DSS in (1) with the following data:

[

−3

0 −2 1

1 0

A=

0.5 0 , Aτ = −2

]

[ −1

0 1 0.5

0 1

] [ ] −0.5 0 −1 , B = 1 , 0.5 0 Fig. 4. Curve of the controller.

[

C = 0

1

]

F = 0.2, M = 0.1

[

[

0

0.1

]T

[ ] , E = 0.2 0 0 , ]T [ ] 0.1 , N = Nτ = 0.2 0 0.2 ,

0 , Dw = 0

0

J(t) =[ 0.4sint , H = 0.5. The parametric matrix K is selected as ] K = −3.75 −5.75 −4.25 for design purpose. The matching uncertain function g(t , x), diffusion function h(t , x), observer perturbation ∆L(t) and external disturbance w (t) are introduced as g(t , x) =

[√

2cos(t)

0.5sin(t)

1 − sin(t) x(t),

]

0 0 0

[ h(t , x) =

0.4 0.2 0.2

0 0 x(t), ∆L(t) = 0

]

0.1sin(t) 0 , 0.1cos(t)

[

]

and w (t) = sin(2t)e−0.2t . In this position, we set X = I3 and Y = 1 such that the equality condition in (11) is satisfied in advance. Given the parameters γ = 1.1245, τ = 0.2 and q = 0.1, the following solutions can

8

Z. Liu, H.R. Karimi and J. Yu / Automatica 111 (2020) 108596

Fig. 5. Curve of the adaptive parameter ρˆ 1 (t).

Fig. 6. Time response of the system state.

Fig. 7. Time response of the state estimation.

Fig. 8. Curve of the linear sliding function.

be obtained in view of conditions (9) and (10): 3.6252 1.2139 −0.0131

] [ ] −0.0131 1.4050 2.3039 , Z = 62.8059 , U1 = 2.2626 0.8041 [ ] 3.8463 0.1872 0.0052 U2 = 0.1872 51.8927 −0.0024 , κ = 7.8703, 0.0052 −0.0024 2.3963 λ1 = 0.7992, λ2 = 0.7796, λ3 = 53.4008. [

1.2139 6.1880 2.3039

The gain matrix L of the observer is obtained as L = 1.4050

[

62.8059

]T

0.8041

.

In this example, the initial conditions are chosen as x(θ ) = [−2.5 0 − 1.5]T , and xˆ (θ ) = [−0.5 1.5 1.0]T , θ ∈ [−0.2, 0]. By the scheme of the proposed sliding surface, state observer and the adaptive switching controller u(t) with σ = 1.8, υ1 = 1.5 and υ2 = 0.75, the simulation results along 11 individual Wiener process paths are shown by Figs. 6–10. It is seen that the desirable property of the system can also be achieved. 5. Conclusions In this paper, a new methodology of the observer-based SMC for uncertain DSS has been proposed. A novel linear sliding surface has been presented based on the designed observer, by which the associated controller has been further developed to

Fig. 9. Curve of the controller.

tackle the unknown uncertainties and external disturbance with the adaptive laws. A new sufficient exponential stability condition with passivity for the closed-loop system during the predefined sliding surface has been derived by employing the stochastic stability theory and linear matrix inequality approach. Lastly, two illustrative examples have been shown to justify the validity of the proposed method via simulation studies. This note is intended

Z. Liu, H.R. Karimi and J. Yu / Automatica 111 (2020) 108596

Fig. 10. Curve of the adaptive parameter ρˆ 1 (t).

to give a complementary approach for the SMO of DSS from both theoretical and technical points. References Aguilar-Lopez, R., & Maya-Yescas, R. (2005). State estimation for nonlinear systems under model uncertainties: a class of sliding-mode observers. Journal of Process Control, 15(3), 363–370. Basin, M., Ferreira, A., & Fridman, L. (2007). Sliding mode identification and control for linear uncertain stochastic systems. International Journal of Systems Science, 38(11), 861–869. Edwards, C., & Spurgeon, S. (1998). Sliding mode control: Theory and applications. Taylor & Francis. Gao, Q., Feng, G., Liu, L., Qiu, J., & Wang, Y. (2014). An ISMC approach to robust stabilization of uncertain stochastic time-delay systems. IEEE Transactions on Industrial Electronics, 61(12), 6986–6994. Huang, L. R., & Mao, X. R. (2010). SMC design for robust H∞ control of uncertain stochastic delay systems. Automatica, 46(2), 405–412. Kao, Y. G., Xie, J., Wang, C. H., & Karimi, H. R. (2015). A sliding mode approach to H∞ non-fragile observer-based control design for uncertain Markovian neutral-type stochastic systems. Automatica, 52, 218–226. Karimi, H. R. (2012). A sliding mode approach to H∞ synchronization of master– slave time-delay systems with Markovian jumping parameters and nonlinear uncertainties. Journal of The Franklin Institute, 349(4), 1480–1496. Karimi, H. R., Maralani, P. J., & Moshiri, B. (2006). Stochastic dynamic output feedback stabilization of uncertain stochastic state-delayed systems. ISA Transactions, 45(2), 201–213. Li, X., & Decarlo, R. (2003). Robust sliding mode control of uncertain time delay systems. International Journal of Control, 76(13), 1296–1305. Li, H. Y., Shi, P., & Yao, D. Y. (2017). Adaptive sliding-mode control of Markov jump nonlinear systems with actuator faults. IEEE Transactions on Automatic Control, 62(4), 1933–1939. Li, H. Y., Shi, P., Yao, D. Y., & Wu, L. G. (2016). Observer-based adaptive sliding mode control for nonlinear Markovian jump systems. Automatica, 64, 133–142. Liang, J. L., Wang, Z. D., & Liu, X. (2010). Robust passivity and passification of stochastic fuzzy time-delay systems. Information Sciences, 180(9), 1725–1737. Lin, C., Chen, B., Shi, P., & Yu, J. P. (2018). Necessary and sufficient conditions of observer-based stabilization for a class of fractional-order descriptor systems. Systems & Control Letters, 112, 31–35. Liu, Z., & Gao, C. C. (2016). A new result on robust H∞ control for uncertain time-delay singular systems via sliding mode control. Complexity, 21(S2), 165–177. Liu, Y. J., Lu, S. M., Tong, S. C., Chen, X. K., Philip Chen, C. L., & Li, D. J. (2018). Adaptive control-based barrier Lyapunov functions for a class of stochastic nonlinear systems with full state constraints. Automatica, 87, 83–93. Liu, M., & Sun, G. S. (2012). Observer-based sliding mode control for Itoˆ stochastic time-delay systems with limited capacity channel. Journal of The Franklin Institute, 349(4), 1602–1616. Mao, X. R. (2007). Stochastic differential equations and applications (2nd ed.). Chichester, U.K.: Horwood.

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Niu, Y. G., & Ho, D. W. C. (2006). Robust observer design for Itoˆ stochastic timedelay systems via sliding mode control. Systems & Control Letters, 55(10), 781–793. Niu, Y. G., Ho, D. W. C., & Lam, J. (2005). Robust integral sliding mode control for uncertain stochastic systems with time-varying delay. Automatica, 41(5), 873–880. Niu, Y. G., Ho, D. W. C., & Wang, X. Y. (2007). Sliding mode control for Itoˆ stochastic systems with Markovian switching. Automatica, 43(10), 1784–1790. Raoufi, R., Marquez, H. J., & Zinober, A. S. I. (2010). H∞ Sliding mode observers for uncertain nonlinear Lipschitz systems with fault estimation synthesis. International Journal of Robust and Nonlinear Control, 20(16), 1785–1801. Rubagotti, M., Estrada, A., Castanos, F., Ferrara, A., & Fridman, L. (2011). Integral sliding mode control for nonlinear systems with matched and unmatched perturbations. IEEE Transactions on Automatic Control, 56(11), 2699–2704. Senthilkumar, T., & Balasubramaniam, P. (2011). Delay-dependent robust H∞ control for uncertain stochastic T-S fuzzy systems with time-varying state and input delays. International Journal of Systems Science, 42(5), 877–887. Shi, P., Liu, M., & Zhang, L. X. (2015). Fault-tolerant sliding-mode-observer synthesis of Markovian jump systems using quantized measurements. IEEE Transactions on Industrial Electronics, 62(9), 5910–5918. Spurgeon, S. (2008). Sliding mode observers: a survey. International Journal of Systems Science, 39(8), 751–764. Wu, L. G., Gao, Y. B., Liu, J. X., & Li, H. Y. (2017). Event-triggered sliding mode control of stochastic systems via output feedback. Automatica, 82, 79–92. Wu, L. G., & Lam, J. (2008). Sliding mode control of switched hybrid systems with time-varying delay. International Journal of Adaptive Control and Signal Processing, 22(10), 909–931. Wu, L. G., & Zheng, W. X. (2009). Passivity-based sliding mode control of uncertain singular time-delay systems. Automatica, 45(9), 2120–2127. Yan, X. G., Spurgeon, S., & Edwards, C. (2010). Sliding mode control for timevarying delayed systems based on a reduced-order observer. Automatica, 46(8), 1354–1362. Yu, J. P., Zhao, L., Yu, H. S., & Lin, C. (2019). Barrier Lyapunov functionsbased command filtered output feedback control for full-state constrained nonlinear systems. Automatica, 105, 71–79. Zhao, L., Yu, J. P., & Lin, C. (2019). Distributed adaptive output consensus tracking of nonlinear multi-agent systems via state observer and command filtered backstepping. Information Sciences, 478, 355–374. Zhao, Z. H., Yu, J. P., Zhao, L., Yu, H. S., & Lin, C. (2018). Adaptive fuzzy control for induction motors stochastic nonlinear systems with input saturation based on command filtering. Information Sciences, 463, 186–195.

Zhen Liu received his M.S. degree in Operational Research and Cybernetics from Ocean University of China, Qingdao, China, in 2013 and the Ph.D. degree in the direction of Control Theory and Applications from Ocean University of China, Qingdao, China, in 2017; and from 2015 to 2017, he was a Joint Ph.D. candidate at the Department of Engineering, Design and Mathematics, University of the West of England, UK, and the College of Engineering, University of Kentucky, USA. Since 2017, he has been working in Qingdao University. His current research interests include sliding mode control, switched hybrid systems and time-delay systems.

Hamid Reza Karimi (M’06–SM’09), born in 1976, received the B.Sc. (First Hons.) degree in power systems from the Sharif University of Technology, Tehran, Iran, in 1998, and the M.Sc. and Ph.D. (First Hons.) degrees in control systems engineering from the University of Tehran, Tehran, in 2001 and 2005, respectively. From 2009 to 2016 he was a full professor of Mechatronics and Control Systems at University of Agder in Norway. Since 2016, he has been a professor of Applied Mechanics with the Department of Mechanical Engineering, Politecnico di Milano, Milan, Italy. His current research interests include control systems and mechatronics with applications to automotive control systems and wind energy. Prof. Karimi is currently the Editor-in-Chief of the Journal of Cyber–Physical Systems, Editor-in-Chief of the Journal of Machines, Editor-in-Chief of the International Journal of Aerospace System Science and Engineering, Editor-in-Chief of the Journal of Designs, Section Editor-in-Chief of the Journal of Electronics, Section Editor-in-Chief of the Journal of Science Progress, Subject Editor for Journal of The Franklin Institute and a Technical Editor or Associate Editor for some international journals, such as the IEEE Transactions on Industrial Electronics, the IEEE Transactions on Neural Networks and Learning Systems,

10

Z. Liu, H.R. Karimi and J. Yu / Automatica 111 (2020) 108596

the IEEE Transactions on Circuits and Systems-I: Regular Papers, the IEEE/ASME Transactions on Mechatronics, the IEEE Transactions on Systems, Man and Cybernetics: Systems, Information Sciences, Neural Networks, IFAC-Mechatronics, International Journal of Robust and Nonlinear Control, Neurocomputing, ISA Transactions, Journal of Control and Decision, Asian Journal of Control, Journal of Systems and Control Engineering, International Journal of Fuzzy Systems. He is a member of Agder Academy of Science and Letters and also a member of the IEEE Technical Committee on Systems with Uncertainty, the Committee on Industrial Cyber–Physical Systems, the IFAC Technical Committee on Mechatronic Systems, the Committee on Robust Control, and the Committee on Automotive Control. Prof. Karimi awarded as the 2016–2018 Web of Science Highly Cited Researcher in Engineering.

Jinpeng Yu received the B.Sc. degree in automation of Qingdao University, Qingdao, China, in 2002, the M.Sc. degree in system engineering of Shandong University, Jinan, China, in 2006 and the Ph.D. degree from the Institute of Complexity Science, Qingdao University, Qingdao, China, in 2011. He is currently a Distinguished Professor at the School of Automation, Qingdao University. He is a recipient of the Shandong Province Taishan Scholar Special Project Fund and Shandong Province Fund for Outstanding Young Scholars. His research interests include electrical energy conversion and motor control, applied nonlinear control and intelligent systems.