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IMA Journal of Mathematical Control and Information
, Volume Advance Article – Sep 6, 2017

20 pages

/lp/ou_press/sliding-mode-control-for-quantized-semi-markovian-switching-systems-ZJKKASwg7s

- Publisher
- Oxford University Press
- Copyright
- © The authors 2017. Published by Oxford University Press on behalf of the Institute of Mathematics and its Applications. All rights reserved.
- ISSN
- 0265-0754
- eISSN
- 1471-6887
- D.O.I.
- 10.1093/imamci/dnx039
- Publisher site
- See Article on Publisher Site

Abstract This article is concerned with exponential stability analysis for semi-Markovian switching systems subject to logarithmic quantization and bounded disturbances via sliding mode control. In order to design a sliding mode surface on quantized outputs, a state observer is utilized to generate the estimation of system states. Then a sliding mode controller is given to stabilize the resulting closed-loop semi-Markovian switching system. Furthermore, it is demonstrated that the proposed sliding mode controller can guarantee the reachability of the addressed sliding surface. Finally, based on cognitive radio communication system, a numerical example is performed to verify the effectiveness of the sliding mode control design technique. 1. Introduction In modern industrial systems, the physical plant, controller and other components are frequently required to be connected over network links, giving rise to the so-called networked control systems. Ever since then, more and more efforts have been devoted to both the stabilization and the control of networked systems (see Yang et al., 2011; Liu et al., 2012; Yang et al., 2014 and references therein). It should be noted that effects of quantization in networked control systems have been taken into consideration to get better system performance in Yang et al. (2011) and Liu et al. (2012). In fact, the study on quantization can date back to 1950s, and the most fundamental question is how much information is required to be communicated by the quantizer for the sake of achieving a certain objective for the closed-loop system. In recent studies, the quantizer is always regarded as an information coder which converts the continuous signal into piecewise continuous signal taking values in a finite set, which is usually employed when the observation and control signals are sent via limited communication channel. However, it is highly desirable to develop more quantization techniques which are needed for the sensor measurements and control commands over networks since the system outputs in the network environment are always required to be quantized before transmission. With the rapid development of network technology, network-induced delays, packet dropout and disorder caused by limited network bandwidth should be taken into account when dealing with the stability analysis and controller synthesis. In order to describe these frequent unpredictable structural changes, the networked control system with Markovian switching has been presented and utilized to model the mode change of the plant or the network-related problems in the past few years. Some remarkable techniques have emerged to deal with the networked control system with Markovian switching (see for example, Liu et al., 2009; Song et al., 2009; Xia et al., 2009; Xiao et al., 2010; Liu et al., 2014; Liu & Xi, 2015, and the references therein). However, in many practical applications, most of the modelling, analysis and design results for Markovian switching systems should be regarded as special cases of semi-Markovian switching systems. Indeed, Markovian switching systems have certain limitations in some senses since the jump time of a Markovian process obeys exponential distribution. But for semi-Markovian switching systems, the transition rate will be time varying instead of constant in Markovian switching systems. Due to the relaxed conditions on the probability distributions, semi-Markovian switching systems have much wider application domain than conventional Markovian switching systems. Unlike a large number of results on Markovian switching systems, there are few works addressing semi-Markovian switching systems, except for Hou et al. (2006), Huang & Shi (2011), Shen et al. (2015), Lee et al. (2015), Liu et al. (2016), Shen et al. (2017) and Wei et al. (2017). Hence, it is of both theoretical merit and practical interest to develop the stability and stabilization problem for networked control system with semi-Markovian switching. On another research front line, sliding mode control has been successfully applied to a wide variety of practical engineering systems due to its advantages, such as good transient performance, the ability to eliminate external disturbances (Yang et al., 2013, 2014) and model uncertainties satisfying the matching condition, convenience to be performed, reduction of the order of the state equation. Recently, research on sliding mode control for Markovian switching systems saw significant progress (Chen et al., 2013; Luan et al., 2013; Zhang et al., 2013; Wu et al., 2014). These references include robust $$H_{\infty}$$ sliding mode control for Markovian switching systems subject to intermittent observations and partially known transition probabilities (Zhang et al., 2013), adaptive sliding mode control for stochastic Markovian switching system with actuator degradation (Chen et al., 2013), asynchronous $$H_{2}/H_{\infty}$$ filtering for discrete-time stochastic Markovian switching system with randomly occurred sensor nonlinearities (Wu et al., 2014) and finite-time stabilization for Markovian switching system with Gaussian transition probabilities (Luan et al., 2013). However, to the best of the authors’ knowledge, there are few results about the sliding mode control problem for networked control systems with semi-Markovian switching, especially in the presence of logarithmic quantization. In fact, when a practical networked control system does not satisfy the so-called memoryless restriction, the widely used Markovian switching scheme would not be applicable. Besides, the theory and experiments suggest that a semi-Markov process captures the stochastic behaviour in the networked control systems more accurately, such as cognitive radio system (Geirhofer et al., 2007) and master–slave system (Liu et al., 2016). Since the semi-Markovian switching has such a good application background, then it is of both necessity and importance for us to investigate the quantized semi-Markovian switching system. For such reasons, by developing a sliding mode control technique, this article is to shorten such a gap between quantized systems with Markovian switching and semi-Markovian switching. Motivated by the above two problems, the purpose of this article is to investigate the problem of sliding mode control for quantized semi-Markovian switching systems with bounded disturbances. A mathematical transformation is presented to deal with the effects of the output quantization, and a Luenberger observer is designed to generate the estimation of system states. Then a sliding mode controller is developed to stabilize the resulting closed-loop semi-Markovian switching system with bounded disturbances. The proposed sliding mode control can guarantee the reachability of the designed sliding mode surface. As a good application, we consider a semi-Markovian switching model over cognitive radio networks. It is assumed that each channel has three modes over cognitive radio links and is characterized by a Weibull semi-Markov process. Finally, a simulation experiment is given to show the effectiveness and applicability of the proposed technique. The main contributions of this article are summarized as follows in three-fold: (i) To establish exponential stability condition for quantized semi-Markovian switching system with bounded disturbances for the first time; (ii) Designing a sliding mode surface on quantized outputs, and utilizing the Luenberger observer to solve the estimation of system states; (iii) Proposing a sliding mode control technique to stabilize the closed-loop semi-Markovian switching system, and providing the reachability analysis. The remainder of this article is organized as follows: Section 2 contains quantized system description, problem formulation and preliminaries; Section 3 presents exponential stability analysis for the quantized semi-Markovian switching system and reachability analysis; Section 4 provides a numerical example to verify the effectiveness of the proposed results; Concluding remarks are given in Section 5. 1.1. Notations The following notations are used throughout the article. The superscripts $$\top$$ and $$-1$$ denote matrix transposition and matrix inverse, respectively. $$[a,b]$$ denotes the closed interval from real number $$a$$ to real number $$b$$ on $$\mathbb{R}$$, where $$a\leq b$$. $$\mathbb{N}^{+}$$ stands for positive integer, $$\mathbb{R}^n$$ denotes the $$n$$ dimensional Euclidean space and $$\mathbb{R}^{m\times n}$$ is the set of all $$m \times n$$ matrices. $$X<Y$$($$X>Y$$), where $$X$$ and $$Y$$ are both symmetric matrices, means that $$X-Y$$ is negative (positive) definite. $$I$$ is the identity matrix with proper dimensions. For a symmetric block matrix, we use $$\star$$ to denote the terms introduced by symmetry. $$\mathscr{E}$$ stands for the mathematical expectation. $$\{r(t), t\geq 0\}$$ stands for a continuous-time semi-Markovian process and $$\Gamma V(x(t),r(t))$$ denotes the infinitesimal generator of stochastic Lyapunov–Krasovskii functional $$V(x(t),r(t))$$. $$\|v\|$$ is the Euclidean norm of vector $$v$$, $$\|v\|=(v^{\top}v)^{\frac{1}{2}}$$, while $$\|A\|$$ is spectral norm of matrix $$A$$, $$\|A\|=[\lambda_{\max}(A^{\top}A)]^{\frac{1}{2}}$$. $$\lambda _{\max (\min)} (A)$$ is the eigenvalue of matrix $$A$$ with maximum(minimum) real part. Matrices, if their dimensions are not explicitly stated, are assumed to have compatible dimensions for algebraic operations. 2. Problem statement and preliminaries 2.1. Quantized semi-Markovian switching system description Given a complete probability space $$\{{\it{\Omega}}, \mathscr{F}, \mathbf{P}\}$$, where $${\it{\Omega}}$$ is the sample space, $$\mathscr{F}$$ is the algebra of events and $$\mathbf{P}$$ is the probability measure defined on $$\mathscr{F}$$. Let $$\{r(t), t\geq 0\}$$ be a continuous-time semi-Markovian process taking values in a finite state space $$S=\{1,2,3,\ldots,N\}$$. The evolution of semi-Markovian process $$r(t)$$ is governed by the following probability transitions: \[P({r(t+h)} = j|r(t) = i) = \left\{{\begin{array}{@{}cc} {{\pi _{ij}(h)}h + o(h)~~~~~~{\rm{}}i \ne j} \\[3pt] {1 + {\pi _{ii}(h)}h + o(h)~~~{\rm{}}i = j,{\rm{}}} \\ \end{array}} \right.\] where $$h>0$$, $$\lim_{h\rightarrow 0}\frac{o(h)}{h}=0$$, $$\pi_{ij}(h)\geq 0(i,j\in S, i\neq j)$$ is the transition rate from mode $$i$$ to $$j$$ and for any state or mode, it satisfies \begin{align*} \pi_{ii}(h)=-\sum_{j=1,j\neq i}^{N}\pi_{ij}(h). \end{align*} Remark 2.1 It should be pointed out that the probability distribution of sojourn time has extended from exponential distribution to Weibull distribution, and the transition rate in semi-Markovian switching will be time varying instead of constant in Markovian switching (Huang & Shi, 2011). In practice, the transition rate $$\pi_{ij}(h)$$ is general bounded by $$\underline{\pi}_{ij}\leq\pi_{ij}(h)\leq\overline{\pi}_{ij}$$, $$\underline{\pi}_{ij}$$ and $$\overline{\pi}_{ij}$$ are real constant scalars. Then $$\pi_{ij}(h)$$ can always be described by $$\pi_{ij}(h)=\pi_{ij}+{\it{\Delta}}\pi_{ij}$$, where $$\pi_{ij}=\frac{1}{2}(\underline{\pi}_{ij}+\overline{\pi}_{ij})$$ and $$|{\it{\Delta}}\pi_{ij}|\leq\omega_{ij}$$ with $$\omega_{ij}=\frac{1}{2}(\overline{\pi}_{ij}-\underline{\pi}_{ij})$$. Remark 2.2 In continuous-time jump linear systems, the sojourn time $$h$$ is a random variable governed by the continuous probability distribution $$F$$. For example, $$F$$ is an exponential distribution in the continuous-time Markovian jump linear systems. Based on probability distribution $$F$$, the transition rate $$\pi_{ij}(h)$$ is the speed that the mode jumps from mode $$i$$ to mode $$j$$. From the memoryless property of the exponential distribution, $$\pi_{ij}(h)\equiv \pi_{ij}$$ is a constant in Markov process, meaning that the jump speed is independent of the history of the stochastic process. But in this article, we consider the transition rate $$\pi_{ij}(h)$$ is time-varying and sojourn time of the stochastic process is non-exponentially distributed, which is often termed as a continuous Weibull semi-Markov process. In this article, the plant is characterized as a semi-Markovian switching system and represented by \begin{align} &\dot{x}(t)=A(r(t))x(t)+B(r(t))[u(t)+w(t)] \nonumber\\ &y(t)=C(r(t))x(t) \nonumber\\ &y_{Q}(t)=Q(y(t)) \nonumber\\ &x(t_0)=x_0, r(t_0)=r_0, \end{align} (1) where $$r(t)$$ is a semi-Markovian process on the complete probability space $$\{{\it{\Omega}}, \mathscr{F}, \mathbf{P}\}$$, $$x(t)\in \mathbb{R}^n$$ represents the state vector, $$u(t)$$ denotes the control input and $$w(t)$$ denotes the uncertainty disturbance input. $$x_0$$, $$r_{0}$$ and $$t_{0}$$ represent the initial state, initial mode and initial time, respectively. $$A(r(t))$$, $$B(r(t))$$ and $$C(r(t))$$ are known mode-dependent constant matrices with appropriate dimensions. $$y(t)\in \mathbb{R}^l$$ is the control output, $$y_{Q}(t)\in \mathbb{R}^l$$ is the quantized output and $$Q(\cdot)=[Q_{1}(\cdot),Q_{2}(\cdot),\cdots,Q_{l}(\cdot)]^{\top}$$ is the logarithmic quantizer, where $$Q_{i}(\cdot)$$ is assumed to be symmetric, that is, $$Q_{i}(y_{i}(t))=-Q_{i}(-y_{i}(t)), i=1,2,\cdots,l.$$ The set of quantized levels of $$Q_{i}(\cdot)$$ is described by $$\mathcal{Q}_{i}=\left\{\pm\theta_{i}^{(j)}|\theta_{i}^{(j)}=(\rho_{i})^{j}\cdot\theta_{i}^{(0)},j=\pm1,\pm2,\cdots\right\}\bigcup\{\pm\theta_{i}^{(0)}\}\bigcup\{0\},$$ where $$0<\rho_{i}<1$$ denotes the quantizer density of the sub-quantizer $$Q_{i}(\cdot)$$, and $$\theta_{i}^{(0)}>0$$ denotes the initial quantization values for the $$i$$th sub-quantizer $$Q_{i}(\cdot)$$. In this article, the associated quantizer $$Q_{i}(\cdot)$$ is defined as follows: \begin{align} Q_{i}(y_{i}(t))= \left\{ \begin{array}{@{}ll} \theta_{i}^{(j)}&\qquad\textrm{when}\,\,\frac{\theta_{i}^{(j)}}{1+\lambda_{i}}<y_{i}(t)\leq\frac{\theta_{i}^{(j)}}{1-\lambda_{i}} \\[6pt] -Q_{i}(-y_{i}(t))&\qquad\textrm{when}\,\,y_{i}(t)<0 \\[3pt] 0&\qquad \textrm{when}\,\,y_{i}(t)=0, \\ \end{array} \right. \end{align} (2) where $$i=1,2,\cdots,l$$, $$j=\pm1,\pm2,\cdots$$ and $$\lambda_{i}=\frac{1-\rho_{i}}{1+\rho_i}, i=1,2,\cdots,l$$ are the quantizer parameters. Define $${\it{\Lambda}}\triangleq {\rm{diag}}\{\lambda_{1},\lambda_{2},\cdots,\lambda_{l}\}$$, and it can be obtained that $$0<{\it{\Lambda}}<I_{l}$$. From (2), the logarithmic quantizer can be characterized by the following scalar sector condition $$(1-\lambda_{i})y_{i}^{2}(t)\leq Q_{i}(y_{i}(t))y_{i}(t)\leq(1+\lambda_{i})y_{i}^{2}(t)$$ which implies that $$[Q(y(t))-(I_{l}-{\it{\Lambda}})y(t)]^{\top}[Q(y(t))-(I_{l}+{\it{\Lambda}})y(t)]\leq 0.$$ Therefore, the quantization function $$Q(\cdot)$$ can be decomposed as follows: \begin{align} Q(y(t))=(I_{l}-{\it{\Lambda}})y(t)+Q_{s}(y(t)), \end{align} (3) where the piecewise function $$Q_{s}:\mathbb{R}^{l}\rightarrow\mathbb{R}^{l}$$ satisfies \begin{align} Q_{s}^{\top}(y(t))[Q_{s}(y(t))-2{\it{\Lambda}} y(t)]\leq0 ~ \textrm{and} ~ Q_{s}(0)=0. \end{align} (4) As a result, substituting (3) into (1) yields \begin{align} y_{Q}(t)=(I_{l}-{\it{\Lambda}})C(r(t))x(t)+Q_{s}(y(t)). \end{align} (5) For notational simplicity, we denote $$A(r(t))$$, $$B(r(t))$$ and $$C(r(t))$$ by $$A_i$$, $$B_i$$ and $$C_i$$ for $$r(t)=i\in S$$. Then the quantized semi-Markovian switching system (1) can be written as follows: \begin{align} &\dot{x}(t)=A_{i}x(t)+B_{i}[u(t)+w(t)] \nonumber\\ &y(t)=C_{i}x(t) \nonumber\\ &y_{Q}(t)=(I_{l}-{\it{\Lambda}})C_{i}x(t)+Q_{s}(y(t)). \end{align} (6) Remark 2.3 Due to the limited transmission capacity of the communication network and some devices in closed-loop systems, the analysis and design of Markovian switching networked control systems with quantization effects are research subjects of great practical and theoretical significance, which have received considerable attention in the past decades (Dong et al., 2011; Wu et al., 2014). However, it should be pointed out that, as a new challenging problem, quantized sliding mode control design for Markovian or semi-Markovian switching systems has attracted little research effort. In fact, the corresponding results of quantized sliding mode control are significant, which will combine the variable-structure control theory and network technologies together to realize possible practical application in modern control systems. Hence, sliding mode control design for a class of quantized semi-Markovian switching systems is investigated in this article. 2.2. Problem formulation and preliminaries For the quantized semi-Markovian switching system (1), the purpose of this article is to design Luenberger observer, which can obtain the accurate states estimation of system (1) by using quantized output measurements $$y_{Q}(t)$$. Then, a sliding mode controller will be synthesized based on the estimation to stabilize the resulting closed-loop system. The Luenberger observer for system (1) is introduced as follows: \begin{align}\label{observer-system} &\dot{\widehat{x}}(t)=A_{i}\widehat{x}\,(t)+B_{i}u(t)+L_{i}(y_{Q}(t)-\widehat{y}(t)), \nonumber\\ &\widehat{y}(t)=C_{i}\widehat{x}\,(t), \end{align} (7) where $$\widehat{x}\,(t)\in \mathbb{R}^{n}$$ is the observer state and $$L_{i}$$ for $$r(t)=i\in S$$ are the observer gain to be designed. Define the error variable $$\delta(t)=x(t)-\widehat{x}\,(t)$$, $$H=I_{l}-{\it{\Lambda}}$$, together with (5), (6) and (7), one can acquire the error dynamics \begin{align}\label{error-dynamic} \dot{\delta}(t)&=A_{i}\delta(t)+B_{i}w(t)-L_{i}\{HC_{i}x(t)+Q_{s}(y(t))-\widehat{y}(t)\} \nonumber\\ &=[A_{i}-L_{i}HC_{i}]\delta(t)+B_{i}w(t)+[L_{i}C_{i}-L_{i}HC_{i}]\widehat{x}\,(t)-L_{i}Q_{s}(y(t)). \end{align} (8) Furthermore, we substitute (5) into (7) and obtain that \begin{align}\label{observer-dynamic} \dot{\widehat{x}}(t)&=A_{i}\widehat{x}\,(t)+B_{i}u(t)+L_{i}[HC_{i}x(t)+Q_{s}(y(t))-C_{i}\widehat{x}\,(t)] \nonumber\\ &=A_{i}\widehat{x}\,(t)+B_{i}u(t)+L_{i}HC_{i}\delta(t)-[L_{i}C_{i}-L_{i}HC_{i}]\,\widehat{x}\,(t)+L_{i}Q_{s}(y(t)). \end{align} (9) It should be mentioned that the stability of the error dynamics (8) is dependent on the estimation state or observer state $$\widehat{x}\,(t)$$. In consideration of this fact, the stability analysis of the error dynamics (8) and the observer system (9) should be taken into account simultaneously. Before proceeding with the main results, we present the following assumption, definition and lemmas, which play an important role in the proof of the main result. Assumption 2.1 The uncertainty disturbance input $$w(t)$$ is unknown but bounded as $$\|w(t)\|\leq\varrho\|x(t)-\hat{x}(t)\|=\varrho\|\delta(t)\|,$$ where $$\varrho>0$$ is a known constant scalar. Remark 2.4 It should be noticed that we have assumed an unknown but bounded disturbance in our article. The main merits of unknown but bounded disturbance can be two-fold: (1) First of all, only the knowledge of a bound on the realization is assumed, and no any statistical properties need to be satisfied. So this form requires the least amount of a priori knowledge of disturbance or noise. (2) Furthermore, the unknown but bounded framework for the disturbance has been used in many different fields and applications, such as mobile robotics, unmanned air vehicles and computer vision. Definition 2.1 The quantized semi-Markovian switching system (1) is exponential stable in mean square sense if there exist $$\alpha\geq0$$ and $$\beta>0$$ such that for any $$r(t)=i\in S$$, $$x_{0}\in \mathbb{R}^{n}$$ and $$t_{0}\in \mathbb{R}^{+}$$ \begin{align*} \mathscr{E}\{\|x(t)\|^{2}\}\leq\alpha\|x(t_{0})\|^{2}\exp\{-\beta(t-t_0)\}. \end{align*} Lemma 2.1 (Mao & Yuan, 2006) Let $$C^{2}(\mathbb{R}^{n}\times\mathbb{N}^{+}; \mathbb{R}^{+})$$ denotes the family of all non-negative functions on $$\mathbb{R}^{n}\times\mathbb{N}^{+}$$ which are continuously twice differentiable, if there exist a function $$V(x(t),r(t))\in C^{2}(\mathbb{R}^{n}\times\mathbb{N}^{+}; \mathbb{R}^{+})$$ and scalar constants $$c_{1}>0$$, $$c_{2}>0$$, $$c_{3}>0$$ for all $$r(t)=i\in S$$ such that $$c_{1}\|x(t)\|^{2}\leq V(x(t),i)\leq c_{2}\|x(t)\|^{2},~~~{\it{\Gamma}} V(x(t),i)\leq-c_{3}\|x(t)\|^{2}$$ then the system is exponential stable in mean square sense. Lemma 2.2 (Svishchuk, 2000) Let $$\tau_1$$ and $$\tau_2$$ be bounded stopping times such that $$0\leq\tau_1\leq\tau_2$$, a.s. If $$V(x(t),r(t))$$ and $${\it{\Gamma}} V(x(t),r(t))$$ are bounded on $$t\in[\tau_1,\tau_2]$$ with probability 1, then the following equality is held \begin{align*} \mathscr{E}\{V(x(\tau_2),r(\tau_2))\}=\mathscr{E}\{V(x(\tau_1),r(\tau_1))\}+\mathscr{E}\int_{\tau_1}^{\tau_2}{\it{\Gamma}} V(x(s),r(s))\,ds. \end{align*} Lemma 2.3 Let $$W\in \mathbb{R}^{n\times n}$$ be a symmetric positive matrix, and let $$x\in \mathbb{R}^n$$, then the following inequality holds $$\lambda_{min}(W)x^{\top}x\leq x^{\top}Wx\leq\lambda_{max}(W)x^{\top}x. $$ Lemma 2.4 (Xiong et al., 2009) Given any scalar $$\chi>0$$ and matrix $$U\in \mathbb{R}^{n\times n}$$, the following inequality $$\chi(U+U^{\top})\leq\chi^{2}V+UV^{-1}U^{\top}$$ holds for any symmetric positive definite matrix $$V\in \mathbb{R}^{n\times n}$$. Lemma 2.5 For any real vectors $$u$$, $$v$$ with appropriate dimensions and symmetric positive matrix $$Q$$ with compatible dimensions, the following inequality holds: \begin{align*} u^{\top}v+v^{\top}u\leq u^{\top}Qu+v^{\top}Q^{-1}v. \end{align*} 3. Main results Define the following integral sliding mode surface function: \begin{align}\label{integral-sliding-function} s(t,i)=G_{i}\,\widehat{x}\,(t)-\int_{0}^{t}G_{i}[A_{i}+B_{i}K_{i}]\,\widehat{x}\,(\theta)d\theta, \end{align} (10) where $$G_{i}$$, $$K_{i}$$ and $$i\in S$$ are coefficient matrices; $$K_{i}$$ is selected to satisfy $$A_{i}+B_{i}K_{i}$$ Hurwitz and $$G_{i}$$ is designed to make $$G_{i}B_{i}$$ non-singular. Then the derivative of $$s(t,i)$$ with respect to $$t$$ can be obtained as follows \begin{align*} \dot{s}(t,i)&=G_{i}\dot{\,\widehat{x}}(t)-G_{i}(A_{i}+B_{i}K_{i})\,\widehat{x}\,(t) \\ &=G_{i}B_{i}u(t)+G_{i}L_{i}HC_{i}\delta(t)-G_{i}[L_{i}C_{i}-L_{i}HC_{i}]\,\widehat{x}\,(t) \\ &\quad+G_{i}L_{i}Q_{s}(y(t))-G_{i}B_{i}K_{i}\,\widehat{x}\,(t). \end{align*} Let $$\dot{s}(t,i)=0$$, then the equivalent control law $$u_{eq}(t,i)$$ is given as follows \begin{align}\label{sliding-law} u_{eq}(t,i)=K_{i}\,\widehat{x}\,(t)+(G_{i}B_{i})^{-1}[(G_{i}L_{i}C_{i}-G_{i}L_{i}HC_{i})\,\widehat{x}\,(t)-G_{i}L_{i}Q_{s}(y(t))-G_{i}L_{i}HC_{i}\delta(t)]. \end{align} (11) Substituting (11) into (9), we acquire the following sliding mode dynamics with respect to observer state $$\,\widehat{x}\,(t)$$, i.e., \begin{align}\label{sliding-mode-dynamic} &\dot{\,\widehat{x}}(t)=[A_{i}+B_{i}K_{i}+B_{i}(G_{i}B_{i})^{-1}(G_{i}L_{i}C_{i}-G_{i}L_{i}HC_{i})-L_{i}C_{i}+L_{i}HC_{i}]\,\widehat{x}\,(t) \nonumber\\ &~~~~~~~~+[L_{i}HC_{i}-B_{i}(G_{i}B_{i})^{-1}G_{i}L_{i}HC_{i}]\delta(t)+[L_{i}-B_{i}(G_{i}B_{i})^{-1}G_{i}L_{i}]Q_{s}(y(t)) \end{align} (12) Remark 3.1 Since sliding mode strategy has been regarded as an important robust control method due to its excellent advantage of strong robustness against external disturbances and structural mode changes, in this article we deeply explore the stability problem of quantized semi-Markovian switching systems via sliding mode control. For quantized semi-Markovian switching systems, how to construct the sliding surface function and how to complete the reachability analysis may be difficult. Hence, sliding surface function design and reachability analysis of the resulting sliding mode dynamics are the main issues to be addressed in our article. Remark 3.2 It should be mentioned that the integral sliding surface, in particular, has one important advantage that is the improvement of the problem of reaching phase, which is the initial period of time that the system has not yet reached the sliding surface and thus is sensitive to any uncertainties or disturbances that jeopardize the system. Integral sliding surface design solves the problem in that the system trajectories start in the sliding surface from the first time instant (Poznyak et al., 2004; Fridman et al., 2005). The function of integral sliding-mode control is now to maintain the system motion on the integral sliding surface despite model uncertainties and external disturbances, although the system state equilibrium has not yet been reached. According to the sliding mode control theory, two steps should be included in designing the sliding mode controller $$u(t)$$ for the quantized semi-Markovian switching system (1), i.e., to establish the sufficient stability condition for the resulting sliding mode dynamics and to provide the reachability of the sliding surface. Both sides will be investigated in the following two subsections, respectively. 3.1. Exponential stability analysis for the quantized semi-Markovian switching system Theorem 3.1 Given real constant scalars $$\underline{\pi}_{ij}$$, $$\overline{\pi}_{ij}$$, $$i,j\in S$$, real positive scalar $$\varrho>0$$ and the quantizer parameters $${\it{\Lambda}}={\rm{diag}}\{\lambda_{1},\lambda_{2},\cdots,\lambda_{l}\}$$, the semi-Markovian switching system (1) under a logarithmic quantizer with sliding mode function (10) is exponentially stable in mean square sense, if the sliding mode parameter matrix $$G_{i}$$ is selected as $$G_{i}=B_{i}^{\top}X_{i},~~i\in S$$ and there exist symmetric positive definite matrices $$X_i\in\mathbb{R}^{n\times n}$$, $$i\in S$$, symmetric positive definite matrices $$V_{ij},i,j\in S,i\neq j$$, and matrices $$Y_i,~i\in S$$ with appropriate dimensions for any real scalar $$\gamma>0$$ such that \begin{align}\label{LMI-1} \left[ \begin{array}{ccc} {\it{\Xi}}_{i}^{9\times9} & \varpi_{1i} & \varpi_{2i} \\ \star & -\mathcal{V}_i & 0 \\ \star & \star & -\mathcal{V}_i \\ \end{array} \right]<0, \end{align} (13) where \begin{align*} {\it{\Xi}}_{i}^{9\times9}=\left[ \begin{array}{ccccccccc} {\it{\Xi}}_{11i} & {\it{\Xi}}_{12i} & {\it{\Xi}}_{13i} & 2X_{i}B_{i} & {\it{\Xi}}_{15i} & {\it{\Xi}}_{16i} & 0 & 0 &0 \\ \star & {\it{\Xi}}_{22i} & {\it{\Xi}}_{23i} & 0 & 0 & 0 & {\it{\Xi}}_{27i} & 0 & {\it{\Xi}}_{29i} \\ \star & \star & -I_{l} & 0 & 0 & 0 & 0 & Y_{i}^{\top} &0 \\ \star & \star & \star & -B_{i}^{\top}X_{i}B_{i} & 0 & 0 & 0 & 0 &0 \\ \star & \star & \star & \star & -X_{i} & 0 & 0 & 0 &0 \\ \star & \star & \star & \star & \star & -X_{i} & 0 & 0 & 0 \\ \star & \star & \star & \star & \star & \star & -X_{i} & 0 &0 \\ \star & \star & \star & \star & \star & \star & \star & -X_{i} & 0 \\ \star & \star & \star & \star & \star & \star & \star & \star & -I \\ \end{array} \right] \end{align*} with \begin{align*} &{\it{\Xi}}_{11i}={\rm{sym}}\{X_{i}A_{i}+X_{i}B_{i}K_{i}+Y_{i}HC_{i}-Y_{i}C_{i}\}+\sum_{j=1}^{N}\pi_{ij}X_{j}+\sum_{j=1,j\neq i}^{N}\frac{\omega_{ij}^{2}}{4}V_{ij} \\ &{\it{\Xi}}_{22i}={\rm{sym}}\{X_{i}A_{i}-Y_{i}HC_{i}\}+\sum_{j=1}^{N}\pi_{ij}X_{j}+\sum_{j=1,j\neq i}^{N}\frac{\omega_{ij}^{2}}{4}V_{ij}+\gamma\varrho^{2}I \\ &{\it{\Xi}}_{12i}=Y_{i}HC_{i}+C_{i}^{\top}Y_{i}^{\top}-C_{i}^{\top}H^{\top}Y_{i}^{\top},~{\it{\Xi}}_{13i}=C_{i}^{\top}{\it{\Lambda}}^{\top}+Y_{i},~{\it{\Xi}}_{16i}=C_{i}^{\top}Y_{i}^{\top} \\ &{\it{\Xi}}_{15i}={\it{\Xi}}_{27i}=C_{i}^{\top}H^{\top}Y_{i}^{\top},~{\it{\Xi}}_{23i}=C_{i}^{\top}{\it{\Lambda}}^{\top}-Y_{i},~{\it{\Xi}}_{29i}=\frac{1}{\sqrt{\gamma}}X_{i}B_{i} \end{align*} \begin{align*} &\varpi_{1i}=\left( \begin{array}{c} X_{i}-X_{1}~\cdots~X_{i}-X_{i-1}~X_{i}-X_{i+1}~\cdots~X_{i}-X_{N}\\ \textbf{0}_{8\times(N-1)} \\ \end{array} \right) \\ &\varpi_{2i}=\left( \begin{array}{c} \textbf{0}_{1\times(N-1)} \\ X_{i}-X_{1}~\cdots~X_{i}-X_{i-1}~X_{i}-X_{i+1}~\cdots~X_{i}-X_{N} \\ \textbf{0}_{7\times(N-1)} \\ \end{array} \right) \\ &\mathcal{V}_i={\rm{diag}}\{V_{i1},V_{i2},\cdots,V_{i(i-1)},V_{i(i+1)},\cdots,V_{iN} \}. \end{align*} Furthermore, the desired observer gain $$L_{i}$$, $$i\in S$$ are designed as $$L_{i}=X_{i}^{-1}Y_{i},~~~i\in S.$$ Proof. In this article, the stochastic Lyapunov–Krasovskii functional is defined as \begin{align}\label{Lyapunov-Function} V(\,\widehat{x}\,(t),\delta(t),i)=\,\widehat{x}^{\top}(t)X_{i}\,\widehat{x}\,(t)+\delta^{\top}(t)X_{i}\delta(t)\triangleq V(\,\widehat{x}\,(t),i)+V(\delta(t),i), \end{align} (14) where $$X_i>0$$, $$r(t)=i\in S$$. By the infinitesimal operator $${\it{\Gamma}}$$, it is held that $${\it{\Gamma}} V(\,\widehat{x}\,(t),\delta(t),i)={\it{\Gamma}} V(\,\widehat{x}\,(t),i)+{\it{\Gamma}} V(\delta(t),i)$$ It should be noticed that the infinitesimal generator of the Lyapunov–Krasovskii functional for the semi-Markovian switching system is different from the one for general Markovian switching system. According to the definition of infinitesimal operator $${\it{\Gamma}}$$, we compute $${\it{\Gamma}} V(\,\widehat{x}\,(t),i)$$ and $${\it{\Gamma}} V(\delta(t),i)$$ along the trajectory of quantized semi-Markovian switching system, respectively. \begin{align}\label{d-Lyapunov-Function} {\it{\Gamma}} V(\,\widehat{x}\,(t),i)=\lim_{{\it{\Delta}} t\rightarrow 0}\frac{\mathscr{E}\{V(\,\widehat{x}\,(t+{\it{\Delta}} t),r(t+{\it{\Delta}} t))|\,\widehat{x}\,(t),r(t)=i\} -V(\,\widehat{x}\,(t),i)}{{\it{\Delta}} t}, \end{align} (15) where $${\it{\Delta}} t$$ is a small positive number. Based on the result of Huang & Shi (2011), it is held that \begin{align}\label{Infinitesimal-1} {\it{\Gamma}} V(\,\widehat{x}\,(t),i)&=2\,\widehat{x}^{\top}(t)X_{i}[A_{i}+B_{i}K_{i}+B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}L_{i}C_{i}- B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}L_{i}HC_{i} \nonumber\\ &\quad +L_{i}HC_{i}-L_{i}C_{i}]\,\widehat{x}\,(t)+\,\widehat{x}^{\top}(t)\left[\sum_{j=1}^{N}\pi_{ij}(h)X_{j}\right]\,\widehat{x}\,(t) \nonumber\\ &\quad +2\,\widehat{x}^{\top}(t)X_{i}[L_{i}HC_{i}-B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}L_{i}HC_{i}]\delta(t) \nonumber\\ &\quad+2\,\widehat{x}^{\top}(t)X_{i}[L_{i}-B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}L_{i}]Q_{s}(y(t)) \end{align} (16) \begin{align}\label{Infinitesimal-2} {\it{\Gamma}} V(\delta(t),i)&=2\delta^{\top}(t)X_{i}[(A_{i}-L_{i}HC_{i})\delta(t)+B_{i}w(t)+(L_{i}C_{i}-L_{i}HC_{i})\,\widehat{x}\,(t)-L_{i}Q_{s}(y(t))] \nonumber\\ &\quad+\delta^{\top}(t)\left[\sum_{j=1}^{N}\pi_{ij}(h)X_{j}\right]\delta(t) \end{align} (17) Considering $$\pi_{ij}(h)=\pi_{ij}+{\it{\Delta}}\pi_{ij}$$, $${\it{\Delta}}\pi_{ii}=-\sum_{j=1,j\neq i}^{N}{\it{\Delta}}\pi_{ij}$$ and employing Lemma 2.4, we have \begin{align} \sum_{j=1}^{N}\pi_{ij}(h)X_j\leq\sum_{j=1}^{N}\pi_{ij}X_j+\sum_{j=1,j\neq i}^{N}\left[\frac{\omega_{ij}^{2}}{4}V_{ij}+(X_j-X_i)V_{ij}^{-1}(X_j-X_i)\right]. \end{align} (18) Consider the term $$2\,\widehat{x}^{\top}(t)[X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}L_{i}HC_{i}]\,\widehat{x}\,(t)$$ in (16). By Lemma 2.5, it follows that \begin{align}\label{tuidao-1} &2\,\widehat{x}^{\top}(t)[X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}L_{i}HC_{i}]\,\widehat{x}\,(t)\leq[\,\widehat{x}^{\top}(t)X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}] \nonumber\\ &~~~~~~~~~~~~~X_{i}^{-1}[X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}\,\widehat{x}\,(t)]+[\,\widehat{x}^{\top}(t)C_{i}^{\top}H^{\top}L_{i}^{\top}]X_{i}[L_{i}HC_{i}\,\widehat{x}\,(t)] \nonumber\\ &=\,\widehat{x}^{\top}(t)X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}\,\widehat{x}\,(t)+[\,\widehat{x}^{\top}(t)C_{i}^{\top}H^{\top}L_{i}^{\top}]X_{i}[L_{i}HC_{i}\,\widehat{x}\,(t)] \end{align} (19) For the same technique, the following inequalities can be obtained \begin{align}\label{tuidao-2} &2\,\widehat{x}^{\top}(t)X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}L_{i}C_{i}\,\widehat{x}\,(t)\leq \,\widehat{x}^{\top}(t)X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}\,\widehat{x}\,(t) \nonumber\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\,\widehat{x}^{\top}(t)C_{i}^{\top}L_{i}^{\top}X_{i}L_{i}C_{i}\,\widehat{x}\,(t) \end{align} (20) \begin{align}\label{tuidao-3} &-2\,\widehat{x}^{\top}(t)X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}L_{i}HC_{i}\delta(t)\leq \,\widehat{x}^{\top}(t)X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}\,\widehat{x}\,(t) \nonumber\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+\delta^{\top}(t)C_{i}^{\top}H^{\top}L_{i}^{\top}X_{i}L_{i}HC_{i}\delta(t) \end{align} (21) \begin{align}\label{tuidao-4} &-2\,\widehat{x}^{\top}(t)X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}L_{i}Q_{s}(y(t))\leq \,\widehat{x}^{\top}(t)X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}\,\widehat{x}\,(t) \nonumber\\ &~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~+Q_{s}^{\top}(y(t))L_{i}^{\top}X_{i}L_{i}Q_{s}(y(t)) \end{align} (22) With Assumption 2.1, we have \begin{align}\label{tuidao-5} 2\delta^{\top}(t)X_{i}B_{i}w(t)\leq\frac{1}{\gamma}\delta^{\top}(t)X_{i}B_{i}B_{i}^{\top}X_{i}\delta(t)+\gamma\varrho^{2}\delta^{\top}(t)\delta(t) \end{align} (23) Substituting (18)–(22) into (16), it can be deduced that \begin{align}\label{Infinitesimal-1-bianxing} &{\it{\Gamma}} V(\,\widehat{x}\,(t),i)\leq\,\widehat{x}^{\top}(t)[{\rm{sym}}\{X_{i}(A_{i}+B_{i}K_{i}+L_{i}HC_{i}-L_{i}C_{i})\}+4X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}+C_{i}^{\top}H^{\top} L_{i}^{\top}X_{i}L_{i}HC_{i} \nonumber\\ &\quad ~~~~~~~~~~~~~~~~~+C_{i}^{\top}L_{i}^{\top}X_{i}L_{i}C_{i}+\sum_{j=1}^{N}\pi_{ij}X_{j}+\sum_{j=1,j\neq i}^{N}\frac{\omega_{ij}^{2}}{4}V_{ij}+\sum_{j=1,j\neq i}^{N}(X_{j}-X_{i})V_{ij}^{-1}(X_{j}-X_{i})]\,\widehat{x}\,(t) \nonumber\\ &\quad ~~~~~~~~~~~~~~~~~+2\,\widehat{x}^{\top}(t)(X_{i}L_{i}HC_{i})\delta(t)+\delta^{\top}(t)(C_{i}^{\top}H^{\top}L_{i}^{\top}X_{i}L_{i}HC_{i})\delta(t) \nonumber\\ &\quad ~~~~~~~~~~~~~~~~~+2\,\widehat{x}^{\top}(t)(X_{i}L_{i})Q_{s}(y(t))+Q_{s}^{\top}(y(t))(L_{i}^{\top}X_{i}L_{i})Q_{s}(y(t)) \end{align} (24) Together with (17), (18) and (23), we can obtain that \begin{align}\label{Infinitesimal-2-bianxing} &{\it{\Gamma}} V(\delta(t),i)\leq\delta^{\top}(t)[{\rm{sym}}\{X_{i}A_{i}-X_{i}L_{i}HC_{i}\}+\sum_{j=1}^{N}\pi_{ij}X_{j}+\sum_{j=1,j\neq i}^{N}\frac{\omega_{ij}^{2}}{4}V_{ij}+\sum_{j=1,j\neq i}^{N}(X_{j}-X_{i})V_{ij}^{-1}(X_{j}-X_{i})]\delta(t) \nonumber\\ &\quad ~~~~~~~~~~~~~~~~~+2\delta^{\top}(t)(X_{i}L_{i}C_{i}-X_{i}L_{i}HC_{i})\,\widehat{x}\,(t)-2\delta^{\top}(t)X_{i}L_{i}Q_{s}(y(t)) \nonumber\\ &\quad ~~~~~~~~~~~~~~~~~+\frac{1}{\gamma}\delta^{\top}(t)X_{i}B_{i}B_{i}^{\top}X_{i}\delta(t)+\gamma\varrho^{2}\delta^{\top}(t)\delta(t) \end{align} (25) Noticing equation (4), i.e., $$Q_{s}^{\top}(y(t))[Q_{s}(y(t))-2{\it{\Lambda}} y(t)]\leq0$$, which is equivalent to \begin{align}\label{qu-lingwai} -Q_{s}^{\top}(y(t))Q_{s}(y(t))+Q_{s}^{\top}(y(t)){\it{\Lambda}} C_{i}[\,\widehat{x}\,(t)+\delta(t)]+[\,\widehat{x}^{\top}(t)+\delta^{\top}(t)]C_{i}^{\top}{\it{\Lambda}}^{\top}Q_{s}(y(t))\geq0 \end{align} (26) According to (24), (25) and (26), $${\it{\Gamma}} V(\,\widehat{x}\,(t),\delta(t),i)$$ can be deduced as follows: \begin{align}\label{d-Lya-zong} &{\it{\Gamma}} V(\,\widehat{x}\,(t),\delta(t),i)\leq\,\widehat{x}^{\top}(t)[{\rm{sym}}\{X_{i}A_{i}+X_{i}B_{i}K_{i}+X_{i}L_{i}HC_{i}-X_{i}L_{i}C_{i}\}+4X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i} \nonumber\\ &~~+C_{i}^{\top}H^{\top}L_{i}^{\top}X_{i}L_{i}HC_{i}+C_{i}^{\top}L_{i}^{\top}X_{i}L_{i}C_{i}+\sum_{j=1}^{N}\pi_{ij}X_{j}+\sum_{j=1,j\neq i}^{N}\frac{\omega_{ij}^{2}}{4}V_{ij}+\sum_{j=1,j\neq i}^{N}(X_{j}-X_{i})V_{ij}^{-1}(X_{j}-X_{i})]\,\widehat{x}\,(t) \nonumber\\ &~~+\delta^{\top}(t)[{\rm{sym}}\{X_{i}A_{i}-X_{i}L_{i}HC_{i}\}+\sum_{j=1}^{N}\pi_{ij}X_{j}+\sum_{j=1,j\neq i}^{N}\frac{\omega_{ij}^{2}}{4}V_{ij}+ \sum_{j=1,j\neq i}^{N}(X_{j}-X_{i})V_{ij}^{-1}(X_{j}-X_{i}) \nonumber\\ &~~+C_{i}^{\top}H^{\top}L_{i}^{\top}X_{i}L_{i}HC_{i}+\frac{1}{\gamma}X_{i}B_{i}B_{i}^{\top}X_{i}+\gamma\varrho^{2}I]\delta(t)+Q_{s}^{\top}(y(t))(L_{i}^{\top}X_{i}L_{i}-I_{l})Q_{s}(y(t)) \nonumber\\ &~~+2\,\widehat{x}^{\top}(t)(C_{i}^{\top}{\it{\Lambda}}^{\top}+X_{i}L_{i})Q_{s}(y(t))+2\,\widehat{x}^{\top}(t)[X_{i}L_{i}HC_{i}+ C_{i}^{\top}L_{i}^{\top}X_{i}^{\top}-C_{i}^{\top}H^{\top}L_{i}^{\top}X_{i}^{\top}]\delta(t) \nonumber\\ &~~+2\delta^{\top}(t)(C_{i}^{\top}{\it{\Lambda}}^{\top}-X_{i}L_{i})Q_{s}(y(t)). \end{align} (27) Define $$\eta(t)=[\,\widehat{x}^{\top}(t)~~\delta^{\top}(t)~~Q_{s}(y(t))]^{\top}$$, then (27) can be rewritten into the following form \begin{align}\label{zuihou-1} {\it{\Gamma}} V(\,\widehat{x}\,(t),\delta(t),i)\leq\eta^{\top}(t){\it{\Phi}}_{i}\eta(t)=\eta^{\top}(t)\left[ \begin{array}{ccc} {\it{\Phi}}_{11i} & {\it{\Phi}}_{12i} & C_{i}^{\top}{\it{\Lambda}}^{\top}+X_{i}L_{i} \\ \star & {\it{\Phi}}_{22i} & C_{i}^{\top}{\it{\Lambda}}^{\top}-X_{i}L_{i} \\ \star & \star & L_{i}^{\top}X_{i}L_{i}-I_{l} \\ \end{array} \right]\eta(t), \end{align} (28) where \begin{align*} &{\it{\Phi}}_{11i}={\rm{sym}}\{X_{i}A_{i}+X_{i}B_{i}K_{i}+X_{i}L_{i}HC_{i}-X_{i}L_{i}C_{i}\}+4X_{i}B_{i}(B_{i}^{\top}X_{i}B_{i})^{-1}B_{i}^{\top}X_{i}+C_{i}^{\top}H^{\top}L_{i}^{\top}X_{i}L_{i}HC_{i} \\ &\quad ~~~~~~~~~~+C_{i}^{\top}L_{i}^{\top}X_{i}L_{i}C_{i}+\sum_{j=1}^{N}\pi_{ij}X_{j}+\sum_{j=1,j\neq i}^{N}\frac{\omega_{ij}^{2}}{4}V_{ij} +\sum_{j=1,j\neq i}^{N}(X_{j}-X_{i})V_{ij}^{-1}(X_{j}-X_{i}), \\ &{\it{\Phi}}_{22i}={\rm{sym}}\{X_{i}A_{i}-X_{i}L_{i}HC_{i}\}+C_{i}^{\top}H^{\top}L_{i}^{\top}X_{i}L_{i}HC_{i}+\sum_{j=1}^{N}\pi_{ij}X_{j}+\sum_{j=1,j\neq i}^{N}\frac{\omega_{ij}^{2}}{4}V_{ij} \\ &\quad ~~~~~~~~~~+\sum_{j=1,j\neq i}^{N}(X_{j}-X_{i})V_{ij}^{-1}(X_{j}-X_{i})+\frac{1}{\gamma}X_{i}B_{i}B_{i}^{\top}X_{i}+\gamma\varrho^{2}I \\ &{\it{\Phi}}_{12i}=X_{i}L_{i}HC_{i}+C_{i}^{\top}L_{i}^{\top}X_{i}^{\top}-C_{i}^{\top}H^{\top}L_{i}^{\top}X_{i}^{\top}. \end{align*} From (28), it is obviously that if $${\it{\Phi}}_{i}<0$$, then $${\it{\Gamma}} V(\,\widehat{x}\,(t),\delta(t),i)<0$$ holds. By choosing $$L_{i}=X_{i}^{-1}Y_{i}$$, and using Schur Complement Lemma, $${\it{\Phi}}_{i}<0$$ is equivalent to that \begin{align}\label{zuihou-2} {\it{\Xi}}_{i}^{9\times9}+{\rm{diag}}\left\{\sum_{j=1,j\neq i}^{N}(X_{j}-X_{i})V_{ij}^{-1}(X_{j}-X_{i}),\sum_{j=1,j\neq i}^{N}(X_{j}-X_{i})V_{ij}^{-1}(X_{j}-X_{i}),0,0,0,0,0,0,0\right\}<0 \end{align} (29) Utilizing Schur Complement Lemma again, it can be concluded that (29) implies to the linear matrix inequality (LMI) condition (13). So we obtain that \begin{align*} {\it{\Gamma}} V(\,\widehat{x}\,(t),\delta(t),i)\leq\eta^{\top}(t){\it{\Phi}}_{i}\eta(t)<0 \end{align*} By Lemma 2.3, which implies to \begin{align*} {\it{\Gamma}} V(\,\widehat{x}\,(t),\delta(t),i)\leq-\min_{i\in S}\lambda_{\min}\{-{\it{\Phi}}_{i}\}\|\eta(t)\|^{2} \end{align*} Then by Lemmas 2.1 and 2.2, and employing the same techniques as in Mao (2002), it can be shown that quantized semi-Markovian switching system (1) under a logarithmic quantizer with sliding mode function (10) is exponentially stable in mean square sense, which completes this proof. □ Remark 3.3 Theorem 3.1 provides a sufficient condition to guarantee exponential stability for the quantized semi-Markovian switching system with observer-based controller. It should be noticed that little attention has been paid to developing numerically testable exponential stability conditions for quantized semi-Markovian switching systems via sliding mode controller, although the stability and control design problems for semi-Markovian switching systems have been receiving increasing interest, which can be found in Johnson (1989), Schwartz (2003), Limnios et al. (2005), Hou et al. (2006), Huang & Shi (2011) and Li et al. (2015) for more details. Remark 3.4 In many engineering applications, the dimension of the matrix may be high and the number of matrix variables may be large in a networked control system. However, equation (13) is LMI and it can be easily calculated by the existing powerful tools, such as LMI toolbox of Matlab. The high dimension and the large number variables can be treated in a short time with a high performance computer. Moreover, if (13) is feasible, then the sliding mode parameter matrix $$G_{i}$$ can be obtained by $$G_{i}=B_{i}^{\top}X_{i}$$. 3.2. Reachability analysis The following theorem demonstrates that the sliding motion can be driven onto the pre-specified sliding surface $$s(t,i)=0$$ in a finite time under the provided sliding mode control law. Theorem 3.2 For Given $$G_{i},~i\in S$$ in the sliding surface function (10) is chosen as $$G_{i}=B_{i}^{\top}X_{i}$$ where $$X_{i}>0,~i\in S$$ are designed as in Theorem 3.1. If the sliding mode control law $$u(t,i)$$ is utilized as follows \begin{align}\label{LMI-2} u(t,i)=-\vartheta s(t,i)+K_{i}\,\widehat{x}\,(t)-\zeta(t,i){\rm{sgn}}(s(t,i)), \end{align} (30) where $$\vartheta>0$$ is a real constant, $${\rm{sgn}}(\cdot)$$ is the familiar sign function, and $$\zeta(t,i)$$ is given as follows: \begin{align}\label{zeta-function} \zeta(t,i)=\max_{i\in S}\{(\|G_{i}L_{i}y_{Q}(t)\|+\|G_{i}L_{i}C_{i}\,\widehat{x}\,(t)\|)\cdot\|(G_{i}B_{i})^{-1}\|\} \end{align} (31) then state trajectories of the closed-loop semi-Markovian switching system (12) will be driven onto the sliding surface $$s(t,i)=0$$ in a finite time and maintained there in the subsequent time. Proof. Select the following Lyapunov function for a mode $$i\in S$$ in this article \begin{align}\label{sliding-Lyapunov-Function} V_{s}(t,i)=\frac{1}{2}s^{\top}(t,i)(B_{i}^{\top}X_{i}B_{i})^{-1}s(t,i). \end{align} (32) According to $$\dot{s}(t,i)=G_{i}B_{i}u(t)+G_{i}L_{i}HC_{i}x(t)-G_{i}L_{i}C_{i}\,\widehat{x}\,(t)+G_{i}L_{i}Q_{s}(y(t))-G_{i}B_{i}K_{i}\,\widehat{x}\,(t).$$ Together with (30), it can be obtained that \begin{align}\label{d-sliding} &\dot{s}(t,i)=G_{i}B_{i}[-\vartheta s(t,i)+K_{i}\,\widehat{x}\,(t)-\zeta(t,i){\rm{sgn}}(s(t,i))]+G_{i}L_{i}HC_{i}x(t) \nonumber\\ &~~~~~~~~~~~~~-G_{i}L_{i}C_{i}\,\widehat{x}\,(t)+G_{i}L_{i}Q_{s}(y(t))-G_{i}B_{i}K_{i}\,\widehat{x}\,(t). \end{align} (33) From (32) and (33), we have \begin{align}\label{d-sliding-Lyapunov-Function} \dot{V}_{s}(t,i)&=s^{\top}(t,i)(GB)^{-1}\dot{s}(t,i) \nonumber\\ &=s^{\top}(t,i)(GB)^{-1}\{G_{i}B_{i}[-\vartheta s(t,i)+K_{i}\,\widehat{x}\,(t)-\zeta(t,i){\rm{sgn}}(s(t,i))] \nonumber\\ &\quad +G_{i}L_{i}HC_{i}x(t)-G_{i}L_{i}C_{i}\,\widehat{x}\,(t)+G_{i}L_{i}Q_{s}(y(t))-G_{i}B_{i}K_{i}\,\widehat{x}\,(t)\}. \end{align} (34) Remembering $$y_{Q}(t)=HC_{i}x(t)+Q_{s}(y(t))$$, it indicates that \begin{align}\label{d-sliding-Lyapunov-Function-guji-0} &\dot{V}_{s}(t,i)\leq -\vartheta \|s(t,i)\|^{2}-s^{\top}(t,i)\zeta(t,i){\rm{sgn}}(s(t,i)) \nonumber\\ &~~~~~~~~~~~~~~+\|s(t,i)\|\cdot\|(G_{i}B_{i})^{-1}\|\cdot(\|G_{i}L_{i}y_{Q}(t)\|+\|G_{i}L_{i}C_{i}\,\widehat{x}\,(t)\|) \end{align} (35) Substituting (31) into (35), due to $$s(t,i){\rm{sgn}}(s(t,i))\triangleq\|s(t,i)\|_{1}\geq\|s(t,i)\|$$, it can be derived that \begin{align}\label{d-sliding-Lyapunov-Function-guji} \dot{V}_{s}(t,i)&\leq-\vartheta\|s(t,i)\|^{2}+(\|s(t,i)\|-\|s(t,i)\|_{1})\cdot\|(G_{i}B_{i})^{-1}\|\cdot (\|G_{i}L_{i}y_{Q}(t)\|+\|G_{i}L_{i}C_{i}\,\widehat{x}\,(t)\|) \nonumber\\ &\leq-\vartheta\|s(t,i)\|^{2} ~~~~\textrm{for}~~s(t,i)\neq0. \end{align} (36) Therefore, we can conclude that the state trajectories of the observer dynamics (9) can be driven onto the sliding surface $$s(t,i)=0$$ by the law (30) in a finite time, which completes this proof. □ Remark 3.5 In Theorem 3.2, if the sliding surface $$s(t,i)$$ converges to zero, it probably jumps to another sliding surface $$s(t,j)$$ at time $$t+{\it{\Delta}} t$$. During the period of $$s(t+{\it{\Delta}} t,j)$$ converging to zero, the condition $$s(t+{\it{\Delta}} t,i)=0$$ is destroyed. So the chattering or jumping effect may be generated along with stochastic process. In order to avoid chattering, we replace the function $${\rm{sgn}}(s(t,i))$$ with $$\frac{s(t,i)}{\varepsilon+\|s(t,i)\|}$$, where $$\varepsilon>0$$ is an adjustable parameter, but how to better reduce chattering or jumping effect will be a research topic in future studies. Remark 3.6 The purpose of our article is to synthesize mode-dependent sliding surface (10) to guarantee the exponential stability of sliding mode dynamics (12) on the surface. In Theorem 3.2, we have constructed a sliding mode scheme (30) such that the state trajectories of the closed-loop system arrive at the sliding surface in finite time. Moreover, similar to the sliding mode control design (Wei et al., 2017), it also can be checked that the mode-independent sliding mode controller synthesis provides no feasible solutions for the semi-Markovian switching system. 4. Numerical examples In this section, we will utilize a semi-Markovian switching model over cognitive radio networks (Shah, 2013; Siraj & Alshebeili, 2013) to illustrate the usefulness and effectiveness of sliding mode control developed in this article. The main attention is focused on designing sliding mode controller for quantized semi-Markovian switching system. Example 4.1 In Ma (2012), it indicates that cognitive radio systems hold promise in the design of large-scale systems due to huge needs of bandwidth during interaction and communication between subsystems. A semi-Markov process has been used to represent the switch between busy and idle states. However, in this article we consider each channel has three states (busy, idle and inactive) and the switch between the modes is governed by a semi-Markov process taking values in $$S=\{1,2,3\}$$. Furthermore, in the cognitive radio structure, it is assumed that at each step the sensor in cognitive radio infrastructure scans only one channel. This assumption avoids the use of costly and complicated multichannel sensors. The sensor first picks a channel to scan, then transmits the signal through it if the channel is idle, or stops transmissions to avoid collision, or stops transmissions for maintenance. In this example, we assume that there are three channels to be sensed and each channel has three modes over cognitive radio links. Moreover, each channel is characterized by a Weibull semi-Markov process. So we can obtain a semi-Markovian switching system which can be described by (1). The system parameters are described as follows: \begin{align*} &A_1 = \begin{bmatrix} -0.5 & 1 & 0\\ 0 & -0.2 & 0\\ 1 & 0 & -1\\ \end{bmatrix}\!,\;\; B_1 = \begin{bmatrix} 1 & -0.5\\ 0 & 0.5\\ 0.6& 1 \\ \end{bmatrix}\!,\;\; C_1 = [1~~0~~1],\\ &A_2 = \begin{bmatrix} -0.8 & 1.5 & 0\\ 0 & -0.1 & 0\\ 1 & 0 & -1.5\\ \end{bmatrix}\!,\;\; B_2 = \begin{bmatrix} 1.5 & -0.5\\ 0 & 0.5\\ 0.6 & 1\\ \end{bmatrix}\!,\;\; C_2 =[1~~1~~2],\\ &A_3 = \begin{bmatrix} -1 & 0.5 & 0\\ 0 & -1 & 0 \\ 0.5 & 0 & -0.5\\ \end{bmatrix}\!,\;\; B_3 = \begin{bmatrix} 1 & -1\\ 0 & -0.5\\ 0.5 & 1\\ \end{bmatrix}\!,\;\; C_3 = [1~~2~~0]. \end{align*} In this article, we assume that the control center (the receiver) has knowledge of the statistics of each channel. According to historical statistics law which is mainly based on Monte-Carlo simulations, so we can obtain bounds of the mode transition rates for semi-Markovian switching over cognitive radio links. As a simulation experiment, the bounds of elements in transition rates matrix are given as follows: \begin{align*} & \pi_{11}(h)\in[-2.08,-1.92],~~\pi_{12}(h)\in[0.88,1.12],~~\pi_{13}(h)\in[0.96,1.04], \\ & \pi_{21}(h)\in[0.95,1.05],~~\pi_{22}(h)\in[-2.11,-1.89],~~\pi_{23}(h)\in[0.94,1.06], \\ & \pi_{31}(h)\in[0.90,1.10],~~\pi_{32}(h)\in[0.92,1.08],~~\pi_{33}(h)\in[-2.06,-1.94]. \end{align*} The semi-Markovian switching signal is displayed in the following Fig. 1, where ‘1’, ‘2’ and ‘3’ correspond to the first, second and third modes, respectively. For the logarithmic quantizer (2), the associated parameters are chosen as $$\rho_{1}=\frac{1}{3}$$, $$\theta_{1}^{(0)}=40$$, and thus $${\it{\Lambda}}=0.2$$, $$H=0.8$$. According to the parameters listed above, we choose $$\gamma=1$$, $$\varrho=0.2$$ and solve the LMI condition (13) in Theorem 3.1, then we obtain that \begin{align*} &L_{1}=\left[ \begin{array}{c} 0.4645 \\ 0.0714 \\ 0.1593 \\ \end{array} \right],~~G_{1}=\left[ \begin{array}{ccc} 0.1683 & -0.0234 & 0.0896 \\ 0.0017 & 0.3099 & -0.0132 \\ \end{array} \right] \\ &L_{2}=\left[ \begin{array}{c} -0.6194 \\ -0.1940 \\ 0.0597 \\ \end{array} \right],~~G_{2}=\left[ \begin{array}{ccc} 0.0443 & 0.0256 & -0.0163 \\ 0.1051 & 0.1109 & 0.2178 \\ \end{array} \right] \\ &L_{3}=\left[ \begin{array}{c} 0.3024 \\ 0.0941 \\ -0.0792 \\ \end{array} \right],~~G_{3}=\left[ \begin{array}{ccc} 0.5473 & -0.9324 & 0.1054 \\ 0.1095 & -1.2330 & -0.1705 \\ \end{array} \right], \\ \end{align*} where $$L_{i}=X_{i}^{-1}Y_{i},~i\in S=\{1,2,3\}$$ are the observer gains and $$G_{i}=B_{i}^{\top}X_{i},~i\in S=\{1,2,3\}$$ are the sliding mode parameter matrices. By the method of Eigenvalue–Eigenvector Placement in system control theory, the following state feedback gains are selected as \begin{align*} &K_1 = \begin{bmatrix} -17.5313 & 119.0167 & -18.5037\\ -45.0302 & -55.7857 & -18.4689 \\ \end{bmatrix}\!, \\ &K_2 = \begin{bmatrix} -18.9845 & 120.5623 & -20.8976\\ -40.0125 & -50.7652 & -18.8796\\ \end{bmatrix}\!, \\ &K_3 = \begin{bmatrix} -18.7613 & 129.7605 & 15.2821\\ 67.5622 & -60.7654 & 20.6754\\ \end{bmatrix}\!, \\ \end{align*} such that $$A_{i}+B_{i}K_{i}$$, $$i\in S=\{1,2,3\}$$ are Hurwitz. We choose $$\vartheta=0.1$$, by Theorem 3.2, the sliding mode controller is designed as (29), (30). The numerical simulation is performed from the initial state $$x(0)=[-0.5,0.5,1]^{\top}$$, and the simulation results are given in the following Figs 2–7. Furthermore, to prevent the control signals from chattering, we replace the function $${\rm{sgn}}(s(t,i))$$ with $$\frac{s(t,i)}{0.1+\|s(t,i)\|},~~i\in S=\{1,2,3\}$$. The trajectories of $$x(t)$$, $$s(t,i)$$ along the cognitive radio system described by quantized semi-Markovian switching system (1) are illustrated in Figs 2 and 3, respectively. Figures 4–6 show the trajectories of control input $$u(t)$$ with mode ‘1’, mode ‘2’ and mode ‘3’, respectively. The trajectories of system output $$y(t)$$ and quantized output $$Q(y(t))$$ are shown in Fig. 7. From these figures, it is demonstrated that the closed-loop quantized semi-Markovian switching system with bounded disturbance is exponentially stable via the sliding mode controller. Fig. 1. View largeDownload slide Switching signal $$r(t)$$ with three modes ‘1’, ‘2’ and ‘3’. Fig. 1. View largeDownload slide Switching signal $$r(t)$$ with three modes ‘1’, ‘2’ and ‘3’. Fig. 2. View largeDownload slide State response of the closed-loop quantized semi-Markovian switching system. Fig. 2. View largeDownload slide State response of the closed-loop quantized semi-Markovian switching system. Fig. 3. View largeDownload slide Sliding mode surface $$s(t,i)$$ with three modes ‘1’, ‘2’ and ‘3’. Fig. 3. View largeDownload slide Sliding mode surface $$s(t,i)$$ with three modes ‘1’, ‘2’ and ‘3’. Fig. 4. View largeDownload slide Control input $$u(t)$$ with mode ‘1’. Fig. 4. View largeDownload slide Control input $$u(t)$$ with mode ‘1’. Fig. 5. View largeDownload slide Control input $$u(t)$$ with mode ‘2’. Fig. 5. View largeDownload slide Control input $$u(t)$$ with mode ‘2’. Fig. 6. View largeDownload slide Control input $$u(t)$$ with mode ‘3’. Fig. 6. View largeDownload slide Control input $$u(t)$$ with mode ‘3’. Fig. 7. View largeDownload slide System output $$y(t)$$ and quantization function $$Q(y(t)).$$ Fig. 7. View largeDownload slide System output $$y(t)$$ and quantization function $$Q(y(t)).$$ 5. Conclusions In this article, the design problem of sliding mode control has been addressed for the semi-Markovian switching system under logarithmic quantization for the first time. A mathematical transformation has been presented to deal with the effect of output quantization. Then a sliding mode controller has been designed to stabilize the closed-loop quantized semi-Markovian switching system, and the reachability analysis is provided to ensure the system’s trajectories to the predefined surface in a finite time. On the basis of cognitive radio communication networks, a numerical example has been given to illustrate the effectiveness of the proposed design schemes. Future work will be focused on quantized semi-Markovian switching networked systems with data dropout and time-varying communication delays or sensor delays. Besides, some other more complex systems with quantized output will be considered in the near future. Funding National Key Scientific Research Project (61233003), in part; and the Fundamental Research Funds for the Central Universities. References Chen, B., Niu, Y. & Zou, Y. 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