# Hybrid-Driven Mechanism Based on Uncertain Network for Markov Jump System with Quantizations and Delay

Authors: Tian Tan; Jinfeng Gao; Jinxia Wang; Zhen Zhao; Miao Ma
Journal: Complexity
 (2020)

---

## Abstract

This paper investigates the hybrid-driven mechanism problem for Markov jump system, where both channel quantization (BCQ) and network-induced delay based on uncertain network are considered. Firstly, comparing with the traditional event-triggered scheme, a hybrid-driven mechanism is employed in networked control systems (NCSs) for the finite capacity of communication bandwidth resources and system performance in equilibrium. Then, the quantization technology is applied in the communication channel from sensor-to-controller and controller-to-actuator. The application of BCQ is for further investigation that mitigate data packet transmission rate. Thirdly, Markov jump system is modeled for the hybrid-driven mechanism and network-induced delay. By constructing the Lyapunov–Krasovskii function, a sufficient condition is derived as the stability criterion, and the controller is designed in which the nonlinear term is rewritten for simplifying the calculation. Finally, two simulation examples are provided to demonstrate the effectiveness of the proposed approach.


---

## Body

## 1. Introduction

Networked control systems (NCSs), distributed systems in which control components such as sensor, controller, and actuator are connected to each other through the communication network channels, are widely applied in various areas such as industrial-control field, multiagent collaboration systems, and smart grids [1]. Accordingly, much attention of researchers is paid to analyzing the NCSs during the past several decades. In contrast with traditional point-to-point data transmission strategy, great advantages of NCSs are in the physical and economic dimension, including low complexity, flexibility, low cost, high reliability, and economic benefits [2]. However, due to the introduction of control objects, a few issues are drawn into the closed-loop control systems. Numerous transmission data occupy limited communication bandwidth which causes consequences, for instance, network-induced delays, data packet dropout, and disorder [3–5]. These drawbacks may lead to a threat of system performance and even result in network paralysis. Therefore, the investigation that how to promote the utilization rate of communication bandwidth is studied extensively in [6–11] and references therein.The stochastic noise and exogenous disturbance affect the stability and performance of the system greatly. To address the imperfections mentioned effectively, researchers model the random variations of system state as Markov chains. In [12], the researchers propose a Markov jump nonlinear system, where they refer to the controller design of adaptive sliding-mode control. Different system research objects can be modeled as Markov jump process, such as sampling process, stochastic time delay, and the combination of guaranteed cost control (GCC) and neural networks in [13–15]. In [16], Dong et al. devise Markov system GCC problem, in which the stochastic quantization is modeled with hidden Markov process. In [17, 18], authors focus on the data packet dropouts problem which is considered as Markov jump process. More systems and research objects which combine with Markov jump system or semi-Markovian jump system are reported in [19–21]. Obviously, as to data packet dropout, network-induced delay, and uncertain system, Markov jump is pretty effective and more intuitive.In order to tackle the difficulties with regard to control systems information transmission and the communication bandwidth resources constrained, researchers propose a method called quantization. The quantization of the signal can be regarded as mapping that continuous signals are converted to a set of finite discrete signals. Then, only the discrete signals can be transmitted through network communication channel and the objective of relieving the control systems communication burden can be achieved. This method is discussed by many researchers in the literature. In [22–24], the authors investigate the NCSs that the control input is quantified. The application areas of quantization technology are diverse under different network frames. For neural network, a logarithmic static and time-invariant quantizer are considered in [25], and quantization is employed in discrete semi-Markov jump network in [26]. The effects of different logarithmic quantizers are investigated in [27], the so-called convex combination and sector bounded. Multidensity logarithmic quantizer is developed in [28] for the communication resource saving. However, the application of quantization will introduce the quantization error in the meantime, which causes the system to become nonlinear. Inevitably, the stability of the system will be reduced. The system performance should be considered and balanced when selecting the quantization parameters.For the purpose of solving the problems mentioned above, event-triggered scheme is put forward and arouses increasing research attention in recent years. For the traditional time-triggered scheme, sampler operates in a fixed period. A lot of meaningless sampled signals are transmitted through a shared communication network. Comparatively, the event-triggered scheme is a new update mode of control signal, of which signals transmission will be executed only if a certain performance indicator is violated. Therefore, more network bandwidth resources are released for other control components, and system performance can be guaranteed. Up to now, many previous investigations are reported. In [29–31], the event-triggered control scheme is applied in neural network filter designing, fault detection, and sliding-mode control. In-depth research studies of network-based systems oriented to different objects are published [32–34], in which multiagent systems and Markov jump systems are involved. To balance and optimize the data transmission referring to bandwidth occupancy rate and the loss of significative information, Liu et al. [35] proposes a novel method called hybrid-driven mechanism which is composed of time-triggered and event-triggered, and Bernoulli distribution is utilized to decide which communication scheme will be chosen. Therefore, many significant literature studies are published. For instance, a H∞ control problem of the nonlinearity system is researched [36], which employs the hybrid-driven mechanism. In [37, 38], NCSs under cyber attacks with hybrid-driven mechanism are developed. To the best knowledge of author, no work has been carried out in the area of hybrid-driven mechanism for the Markov jump system with quantization and delay. This motivates our study.Inspired by the investigation aforementioned, in this paper, we focus on the hybrid-driven mechanism based on uncertain networked for Markov jump system with quantizations and delay. The main contributions are given as follows:(i)
The network state is modeled as the Markov jump system, and the hybrid-driven mechanism is put forward to determining whether the sampling data should be transmitted or not.(ii)
To optimize the dynamic equilibrium between system performance and communication load contrasting with the existing literature [8], the switching rule between time-triggered scheme and event-triggered scheme will be ensured by stochastic Bernoulli distribution.(iii)
Dual-quantizer is applied for the purpose of reducing communication resources wastage further. The network-induced delay will be considered by the method of [6].(iv)
Via Lyapunov–Krasovskii and free-weighting matrix, a sufficient condition is presented and the desired controller will be designed. A numerical example and a practical example are provided to illustrate the effectiveness of the approach proposed in this paper.The remainder of this paper is organized as follows. In Section2, the problem is narrated and analyzed in detail. The main results of system stability and controller design are derived in Section 3. In Section 4, the results of simulation examples are given. The conclusion is presented in Section 5.


### 1.1. Notation

ℝ
m represents the m-dimensional Euclidean space and ℝm×n means the set of m×n dimension real matrices. ℕ denotes the set of nonnegative integers. I denotes the identity matrix of appropriate dimension. Superscript T is the transposition of matrix and diag … stands for the block diagonal matrix. symA stands for the A+AT. In symmetric matrices, “∗” denotes a term induced by symmetry. The notation P>0<0 stands for real symmetric positive definite.


## 1.1. Notation

ℝ
m represents the m-dimensional Euclidean space and ℝm×n means the set of m×n dimension real matrices. ℕ denotes the set of nonnegative integers. I denotes the identity matrix of appropriate dimension. Superscript T is the transposition of matrix and diag … stands for the block diagonal matrix. symA stands for the A+AT. In symmetric matrices, “∗” denotes a term induced by symmetry. The notation P>0<0 stands for real symmetric positive definite.


## 2. Problem Formulation

### 2.1. System Description

The structure of NCSs is shown in Figure1. Hybrid-driven communication mechanism is the key element for reducing the signal data transmission rate in this system. Consider the Markov jump uncertain system as follows:(1)x˙t=Art+ΔArtxt+Brtut,where xt∈ℝn is the system state vector, ut∈ℝm is the system control input, Art and Brt represent real constant system matrices with appropriate dimensions, and ΔArt is unknown item. It denotes time-varying matrices that are bounded parameter uncertainties, and follow the constraint [39] as follows:(2)ΔArt=DrtFtHrt,where Drt and Hrt are constant matrices with appropriate dimensions. Ft is the unknown time-varying matrix and has the following property:(3)FTtFt≤I,rt is assumed to be the system mode that is a homogeneous Markov jump stochastic process, which obtain values in a finite set S=1,2,…,M. Π=λiji,j∈S denotes the transition probability matrix, and is given by(4)Prt+Δh=jrt=i=λijΔh+oΔh,i≠j,1+λijΔh+oΔh,i=j,where λij represents the transition rate that system jumps from mode i to mode j and follows the principles of limΔh⟶∞oΔh/Δh=0, Δh>0, and λij≥0, λii=−∑j=1,j≠iMλij for i≠j.Figure 1
The structure of the Markov jump system with hybrid-driven mechanism.The transition probability matrix is defined as follows:(5)Π=λ11⋯λ1M⋮⋱⋮λM1⋯λMM.The following assumptions and simplicity is considered in the paper for further development and obtaining the main results:(i)
The loss and disorder of data packet are not considered in the system. A single packet is transmitted at each time-stamped in the network channel. The external disturbance is omitted.(ii)
The logic zero order hold (ZOH) in the actuator is used to hold the control input. In other words, the actuator maintains the last transmitted data until the current sampling data arrive via network.(iii)
For simplicity, we substituteArt, ΔArt, Brt, Drt, and Hrt with Ai, ΔAi, Bi, Di, and Hi, while rt=ii∈S.In this paper, the hybrid-driven mechanism problem for the Markov jump system with quantization and network-induced delay based on uncertain network is developed. The feedback controller expression is given as follows:(6)ut=Kixtkh,t∈tkh+τtk,tk+1h+τtk+1.Similar to [6], τtk is the network-induced communication delay which represents the delayed time with sensor-to-controller and controller-to-actuator. tkh is the sampling instant which satisfy t1,t2,…⊂1,2,3,…. Ki is the controller gain to be designed.Remark 1.
The actuator, quantizers, and controllers are driven by event, and the set of sampling sequences is expressed astkhtk∈ℕ while the sensors and the sampler are time-triggering with a constant sampling period h. The set of data transmitted successfully is described by khk∈ℕ.


### 2.2. Hybrid-Driven Mechanism

In order to reduce the network channel communication burden and save the network resource, comparing with the traditional time-triggered communication scheme and event-triggered communication scheme, inspired by [35], we address the controller problem with hybrid-driven mechanism, which is composed of time-triggered and event-triggered. For the balance of system performance and data transmission frequency, a switching following Bernoulli distribution is used to choose the appropriate communication scheme between two mechanisms.If the hybrid-driven mechanism is under time-triggering, we define network-induced delaydt=t−tkh, where t∈tkh+τtk,tk+1h+τtk+1; then equation, (6) can be rewritten as(7)ut=Kixt−dt,where 0≤dm≤dt≤dM, and dM is the upper bound of the network-induced communication delay.For the purpose of reducing data packet transfer rate, inspired by [22], whether the latest sampling data can be transmitted via the communication channels to the controller and actuator will be decided by the condition as follows:(8)ekTtΦiekt<σxTt−τtΦixt−τt,where ekt=xtkh−xtkh+lh indicates the system error vector, σ∈0,1 is a given scalar triggered parameter, and Φi>0i=1,2 mean symmetric positive triggered matrices. xtkh represents the last transmission data. xtkh+lh is the current sampled signals.Defineρ=tk+1−tk−1, the interval tkh+τtk,tk+1h+τtk+1 are divided into subintervals as tkh+τtk,tk+1h+τtk+1=∪l=0ρΩl, where Ωl=ikl+τtk+1,ikl+h+τtk+l+1, and iklh=tkh+lh, l=0,1,2,…,ρ. Define τt=t−iklh, which has 0≤τm≤τt≤h+τtk+1≜τM. Then, we can get xtkh=xt−τt+ekt, and equation (6) can be rewritten as(9)ut=Kixt−τt+ekt.According to [35], for the switching between time-triggered scheme and event-triggered scheme, Bernoulli distribution is considered to construct hybrid-driven mechanism based on equations (7) and (9). The plant input ut can be rewritten as(10)ut=αtKixt−dt+1−αtKixt−τt+ekt,where αt∈0,1. αt is a stochastic variable. αt=1 means that the NCSs are under the time-driven scheme and the sampling signals are transmitted periodically. αt=0 means that the NCSs are in motion under the event-driven scheme, and the period sampling signals transmitted will be constrained unless the condition of event-triggered scheme is violated.According to the Bernoulli distribution, the mathematical expectation ofαt satisfy Eαt=α¯, and its mathematical variance μ2=α¯1−α¯.Remark 2.
The traditional event-triggered scheme is considered in hybrid-driven mechanism of this paper. More investigation about event-triggered scheme is developed in many literature studies, such as adaptive event-triggered scheme [34, 40]. One possible future research can be directed to employ the adaptive event-triggered scheme into hybrid-driven mechanism.


### 2.3. Quantizer

Based on the limited transmission capability and the aim to decrease the data transmission of the communication channel, for the feature that the network communication quality is seriously affected with high quantization density, while important data packet information will be lost with low quantization density, BCQ is employed in system. As shown in Figure1, the control signal xtkh is converted into hxtkh by the quantizer h⋅. Inspired by [41], at the side of the sensor, we define the quantizer as hx=h1x1h2x2h3x3…hnxnT, where the logarithmic quantizers hixii=1,2,…,n are given by(11)hixi=ωli,ifωli1+δi<xi<ωli1−δi,xi>0,0,ifxi=0,−hi−xi,ifxi<0,where(12)δhi=1−ρhi1+ρhi,ρhi∈0,1, and ρhi denotes the quantization density that is a given constant. Similar to [42], the quantization levels can be described as(13)W=±ωli:ωli=ρhilω0i,l=±1,±2,…∪±ω0i∪0,where ω0i>0. To our best knowledge, the current static quantizer can be summarized as a class of quantizers, of which sector bound condition as the following should be satisfied:(14)Δq≤δu+1−δd,where quantizer design parameters δ∈0,1 and d>0. u represents the input of the quantizer. qu is the output of quantizer and Δq=qu−u denotes the error of quantization. Without loss of generality, we define Δh=diagΔh1,Δh1,…Δhn, where Δhi∈−δhi,δhi.The quantizerhx can be expressed as(15)hx=I+Δhx.Similarly, at the side of the actuator, we define the quantizer asgx=g1x1g2x2g3x3…gnxnT, where gixii=1,2,…,n. Correspondingly, the quantizer gx is expressed as(16)gx=I+Δgx,where Δgi∈−δi,δi, and(17)δgi=1−ρgi1+ρgi,where ρgi is the quantization density.Remark 3.
Static quantization is adopted in this paper for the advantages of the positive mathematical model and convenient design. By contrast, the dynamic quantizer which is composed of the dynamic parameter and the static quantizer is potential on practical stability and the corresponding research is published in [43, 44].
Based on the abovementioned description, at the side of the sensor, the output of quantizerhx can be expressed as(18)ut=αtKiI+Δhxt−dt+1−αtKiI+Δhxt−τt+ekt.
Then, at the side of actuator, the output of quantizergx can be expressed as(19)ut=αtKiI+ΔhI+Δgxt−dt+1−αtKiI+ΔhI+Δg×xt−τt+ekt=αtKi+ΔKixt−dt+1−αtKi+ΔKixt−τt+ekt,where ΔKi=ΔgKi+KiΔh+ΔgKiΔh. Combining (1) with (19), the closed-loop network system can be described as(20)x˙t=Ai+ΔAixt+BiαtKi+ΔKi×xt−dt+Bi1−αtKi+ΔKi×xt−τt+ekt,t∈Ωl,xt=ϕt,t∈−z,0,where ϕt is the initial condition of xt, and z=maxdM,τM.
As to simplify the derivation and expression, the following lemmas are employed.Definition 1.
The state feedback closed-loop Markov jump system (1) which considers the hybrid-driven mechanism is asymptotically stable if there exists a controller ut.Lemma 1 (see [39]).
Given matrices with appropriate dimensionsQ=QT,H,E, then(21)Q+HFtE+ETFTtHT<0,where the necessary and sufficient condition for any Ft satisfying FTFT≤I is that there exists a positive parameter such that(22)Q+ε−1HHT+εETE<0.Lemma 2 (see [45]).
Suppose appropriate dimensional matricesΞ1, Ξ2, Ω and 0<τm≤τt≤τM, the necessary and sufficient condition that made the following inequation(23)τt−τmΞ1+τM−τtΞ2+Ω<0,is(24)τM−τmΞ1+Ω<0,τM−τmΞ2+Ω<0.Lemma 3 (see [46]).
For any given vectorsx, y∈ℝn, the positive definite Q∈ℝn×n satisfy the inequation(25)2xTy≤xTQx+yTQ−1y.Lemma 4 (see [47]).
For matricesR>0, X and any real parameter ρ, the inequality holds(26)−XR−1X≤ρ2R−2ρX.


## 2.1. System Description

The structure of NCSs is shown in Figure1. Hybrid-driven communication mechanism is the key element for reducing the signal data transmission rate in this system. Consider the Markov jump uncertain system as follows:(1)x˙t=Art+ΔArtxt+Brtut,where xt∈ℝn is the system state vector, ut∈ℝm is the system control input, Art and Brt represent real constant system matrices with appropriate dimensions, and ΔArt is unknown item. It denotes time-varying matrices that are bounded parameter uncertainties, and follow the constraint [39] as follows:(2)ΔArt=DrtFtHrt,where Drt and Hrt are constant matrices with appropriate dimensions. Ft is the unknown time-varying matrix and has the following property:(3)FTtFt≤I,rt is assumed to be the system mode that is a homogeneous Markov jump stochastic process, which obtain values in a finite set S=1,2,…,M. Π=λiji,j∈S denotes the transition probability matrix, and is given by(4)Prt+Δh=jrt=i=λijΔh+oΔh,i≠j,1+λijΔh+oΔh,i=j,where λij represents the transition rate that system jumps from mode i to mode j and follows the principles of limΔh⟶∞oΔh/Δh=0, Δh>0, and λij≥0, λii=−∑j=1,j≠iMλij for i≠j.Figure 1
The structure of the Markov jump system with hybrid-driven mechanism.The transition probability matrix is defined as follows:(5)Π=λ11⋯λ1M⋮⋱⋮λM1⋯λMM.The following assumptions and simplicity is considered in the paper for further development and obtaining the main results:(i)
The loss and disorder of data packet are not considered in the system. A single packet is transmitted at each time-stamped in the network channel. The external disturbance is omitted.(ii)
The logic zero order hold (ZOH) in the actuator is used to hold the control input. In other words, the actuator maintains the last transmitted data until the current sampling data arrive via network.(iii)
For simplicity, we substituteArt, ΔArt, Brt, Drt, and Hrt with Ai, ΔAi, Bi, Di, and Hi, while rt=ii∈S.In this paper, the hybrid-driven mechanism problem for the Markov jump system with quantization and network-induced delay based on uncertain network is developed. The feedback controller expression is given as follows:(6)ut=Kixtkh,t∈tkh+τtk,tk+1h+τtk+1.Similar to [6], τtk is the network-induced communication delay which represents the delayed time with sensor-to-controller and controller-to-actuator. tkh is the sampling instant which satisfy t1,t2,…⊂1,2,3,…. Ki is the controller gain to be designed.Remark 1.
The actuator, quantizers, and controllers are driven by event, and the set of sampling sequences is expressed astkhtk∈ℕ while the sensors and the sampler are time-triggering with a constant sampling period h. The set of data transmitted successfully is described by khk∈ℕ.


## 2.2. Hybrid-Driven Mechanism

In order to reduce the network channel communication burden and save the network resource, comparing with the traditional time-triggered communication scheme and event-triggered communication scheme, inspired by [35], we address the controller problem with hybrid-driven mechanism, which is composed of time-triggered and event-triggered. For the balance of system performance and data transmission frequency, a switching following Bernoulli distribution is used to choose the appropriate communication scheme between two mechanisms.If the hybrid-driven mechanism is under time-triggering, we define network-induced delaydt=t−tkh, where t∈tkh+τtk,tk+1h+τtk+1; then equation, (6) can be rewritten as(7)ut=Kixt−dt,where 0≤dm≤dt≤dM, and dM is the upper bound of the network-induced communication delay.For the purpose of reducing data packet transfer rate, inspired by [22], whether the latest sampling data can be transmitted via the communication channels to the controller and actuator will be decided by the condition as follows:(8)ekTtΦiekt<σxTt−τtΦixt−τt,where ekt=xtkh−xtkh+lh indicates the system error vector, σ∈0,1 is a given scalar triggered parameter, and Φi>0i=1,2 mean symmetric positive triggered matrices. xtkh represents the last transmission data. xtkh+lh is the current sampled signals.Defineρ=tk+1−tk−1, the interval tkh+τtk,tk+1h+τtk+1 are divided into subintervals as tkh+τtk,tk+1h+τtk+1=∪l=0ρΩl, where Ωl=ikl+τtk+1,ikl+h+τtk+l+1, and iklh=tkh+lh, l=0,1,2,…,ρ. Define τt=t−iklh, which has 0≤τm≤τt≤h+τtk+1≜τM. Then, we can get xtkh=xt−τt+ekt, and equation (6) can be rewritten as(9)ut=Kixt−τt+ekt.According to [35], for the switching between time-triggered scheme and event-triggered scheme, Bernoulli distribution is considered to construct hybrid-driven mechanism based on equations (7) and (9). The plant input ut can be rewritten as(10)ut=αtKixt−dt+1−αtKixt−τt+ekt,where αt∈0,1. αt is a stochastic variable. αt=1 means that the NCSs are under the time-driven scheme and the sampling signals are transmitted periodically. αt=0 means that the NCSs are in motion under the event-driven scheme, and the period sampling signals transmitted will be constrained unless the condition of event-triggered scheme is violated.According to the Bernoulli distribution, the mathematical expectation ofαt satisfy Eαt=α¯, and its mathematical variance μ2=α¯1−α¯.Remark 2.
The traditional event-triggered scheme is considered in hybrid-driven mechanism of this paper. More investigation about event-triggered scheme is developed in many literature studies, such as adaptive event-triggered scheme [34, 40]. One possible future research can be directed to employ the adaptive event-triggered scheme into hybrid-driven mechanism.


## 2.3. Quantizer

Based on the limited transmission capability and the aim to decrease the data transmission of the communication channel, for the feature that the network communication quality is seriously affected with high quantization density, while important data packet information will be lost with low quantization density, BCQ is employed in system. As shown in Figure1, the control signal xtkh is converted into hxtkh by the quantizer h⋅. Inspired by [41], at the side of the sensor, we define the quantizer as hx=h1x1h2x2h3x3…hnxnT, where the logarithmic quantizers hixii=1,2,…,n are given by(11)hixi=ωli,ifωli1+δi<xi<ωli1−δi,xi>0,0,ifxi=0,−hi−xi,ifxi<0,where(12)δhi=1−ρhi1+ρhi,ρhi∈0,1, and ρhi denotes the quantization density that is a given constant. Similar to [42], the quantization levels can be described as(13)W=±ωli:ωli=ρhilω0i,l=±1,±2,…∪±ω0i∪0,where ω0i>0. To our best knowledge, the current static quantizer can be summarized as a class of quantizers, of which sector bound condition as the following should be satisfied:(14)Δq≤δu+1−δd,where quantizer design parameters δ∈0,1 and d>0. u represents the input of the quantizer. qu is the output of quantizer and Δq=qu−u denotes the error of quantization. Without loss of generality, we define Δh=diagΔh1,Δh1,…Δhn, where Δhi∈−δhi,δhi.The quantizerhx can be expressed as(15)hx=I+Δhx.Similarly, at the side of the actuator, we define the quantizer asgx=g1x1g2x2g3x3…gnxnT, where gixii=1,2,…,n. Correspondingly, the quantizer gx is expressed as(16)gx=I+Δgx,where Δgi∈−δi,δi, and(17)δgi=1−ρgi1+ρgi,where ρgi is the quantization density.Remark 3.
Static quantization is adopted in this paper for the advantages of the positive mathematical model and convenient design. By contrast, the dynamic quantizer which is composed of the dynamic parameter and the static quantizer is potential on practical stability and the corresponding research is published in [43, 44].
Based on the abovementioned description, at the side of the sensor, the output of quantizerhx can be expressed as(18)ut=αtKiI+Δhxt−dt+1−αtKiI+Δhxt−τt+ekt.
Then, at the side of actuator, the output of quantizergx can be expressed as(19)ut=αtKiI+ΔhI+Δgxt−dt+1−αtKiI+ΔhI+Δg×xt−τt+ekt=αtKi+ΔKixt−dt+1−αtKi+ΔKixt−τt+ekt,where ΔKi=ΔgKi+KiΔh+ΔgKiΔh. Combining (1) with (19), the closed-loop network system can be described as(20)x˙t=Ai+ΔAixt+BiαtKi+ΔKi×xt−dt+Bi1−αtKi+ΔKi×xt−τt+ekt,t∈Ωl,xt=ϕt,t∈−z,0,where ϕt is the initial condition of xt, and z=maxdM,τM.
As to simplify the derivation and expression, the following lemmas are employed.Definition 1.
The state feedback closed-loop Markov jump system (1) which considers the hybrid-driven mechanism is asymptotically stable if there exists a controller ut.Lemma 1 (see [39]).
Given matrices with appropriate dimensionsQ=QT,H,E, then(21)Q+HFtE+ETFTtHT<0,where the necessary and sufficient condition for any Ft satisfying FTFT≤I is that there exists a positive parameter such that(22)Q+ε−1HHT+εETE<0.Lemma 2 (see [45]).
Suppose appropriate dimensional matricesΞ1, Ξ2, Ω and 0<τm≤τt≤τM, the necessary and sufficient condition that made the following inequation(23)τt−τmΞ1+τM−τtΞ2+Ω<0,is(24)τM−τmΞ1+Ω<0,τM−τmΞ2+Ω<0.Lemma 3 (see [46]).
For any given vectorsx, y∈ℝn, the positive definite Q∈ℝn×n satisfy the inequation(25)2xTy≤xTQx+yTQ−1y.Lemma 4 (see [47]).
For matricesR>0, X and any real parameter ρ, the inequality holds(26)−XR−1X≤ρ2R−2ρX.


## 3. Main Results

In this section, we devise the problem of the Markov jump continuous system based on uncertain network, and the result is given for the stability of the system. The controller is designed with Lyapunov–Krasovskii stability theory.Theorem 1.
For given scalarsτM, dM, 0<σ<1 and probability parameter α¯, gain matrix Ki, the closed-loop system (20) is asymptotically stable under the hybrid-driven mechanism, if there exist matrices Pi>0, Φi>0i=1,2, Q1>0, Q2>0, R1>0, R2>0, and W, V, M, and N with appropriate dimensions such that inequality (27) satisfying the following LMI:(27)Σ11∗∗∗Σ21Σ22∗∗Σ310Σ33∗Σ4100Σ44<0,where(28)Σ11=Θ+ϒ+ϒT,Θ=Θ11∗∗∗∗∗Θ210∗∗∗∗00−Q1∗∗∗Θ4100σΦi∗∗0000−Q2∗Θ610000−Φi,Θ11=AiTPi+PiAi+∑j=1MλijPj+Q1+Q2,Θ21=α¯Ki+ΔKiTBiTPi,Θ41=Θ61=1−α¯Ki+ΔKiTBiTPi,ϒ=M+WN−M−NV−W−V0,Σ21=Ξ1Ξ20Ξ30Ξ4,Ξ1=τMPiAidMPiAi,Ξ2=τMα¯PiBiKi+ΔKidMα¯PiBiKi+ΔKi,Ξ3=Ξ4=τM1−α¯PiBiKi+ΔKidM1−α¯PiBiKi+ΔKi,Ξ5=Ξ6=Ξ7=τMμPiBiKi+ΔKidMμPiBiKi+ΔKi,Σ31=0Ξ50Ξ60Ξ7,Σ41=τMWτMVdMMdMN,Σ22=Σ33=diag−τMPiR1−1Pi,−dMPiR2−1Pi,Σ44=diag−τMR1,−τMR1,−dMR2,−dMR2.Proof.
Considering the closed-loop system (20), we construct the following Lyapunov–Krasovskii function:(29)Vt=V1t+V2t+V3t,where(30)V1t=xTtPixt,V2t=∫t−dMtxTsQ1xsds+∫t−τMtxTsQ2xsds,V3t=∫t−τMt∫t+θtx˙TsR1x˙sdsdθ+∫t−dMt∫t+θtx˙TsR2x˙sdsdθ.
Taking the time-derivative ofVt, then we have(31)V˙t=V˙1t+V˙2t+V˙3t,where(32)V˙1t=x˙TtPixt+xTtPix˙t+xTt∑j=1MλijPjxt,V˙2t=xTtQ1xt−xTt−dMQ1xt−dM+xTtQ2xt−xTt−τMQ2xt−τM,V˙3t=τMx˙TtR1x˙t−∫t−τMtx˙TsR1x˙sds+dMx˙TtR2x˙t−∫t−dMtx˙TsR2x˙sds.
Inspired by the free-weighting matrices, we define(33)2ξTtWxt−xt−τt−∫t−τttx˙sds=0,(34)2ξTtV×xt−τt−xt−τM−∫t−τMt−τtx˙sds=0,(35)2ξTtMxt−xt−dt−∫t−dttx˙sds=0,(36)2ξTtN×xt−dt−xt−dM−∫t−dMt−dtx˙sds=0,where(37)ξTt=ξ1Ttξ2Tt,ξ1Tt=xTtxTt−dtxTt−dM,ξ2Tt=xTt−τtxTt−τMekTt,W=W1TW2TW3TW4TW5TW6TT,V=V1TV2TV3TV4TV5TV6TT,M=M1TM2TM3TM4TM5TM6TT,N=N1TN2TN3TN4TN5TN6TT.
By Lemma2, we can obtain(38)−2ξTtW∫t−τttx˙sds≤τtξTtWR1−1WTξt+∫t−τttx˙TsR1x˙sds,(39)−2ξTtV∫t−τMt−τtx˙sds≤τM−τtξTtVR1−1VTξt+∫t−τMt−τtx˙TsR1x˙sds,(40)−2ξTtM∫t−dttx˙sds≤dtξTtMR2−1MTξt+∫t−dttx˙TsR2x˙sds,(41)−2ξTtN∫t−dMt−dtx˙sds≤dM−dtξTtNR2−1NTξt+∫t−dMt−dtx˙TsR2x˙sds.
Combining (8), (20)–(40), and Lemma 3, we obtain following inequation with taking mathematical expectation into account:(42)EV˙t≤Ex˙TtPixt+xTtPix˙t+xTt∑j=1MλijPjxt+xTtQ1xt−xTt−dMQ1xt−dM+xTtQ2xt−xTt−τMQ2xt−τM+Ex˙TtτMR1+dMR2x˙t+2ξTtWxt−2ξTtWxt−τt+2ξTtVxt−τt−2ξTtVxt−τM+τtξTtWR1−1WTξt+τM−τtξTtVR1−1VTξt+2ξTtMxt−2ξTtMxt−dt+2ξTtNxt−dt−2ξTtNxt−dM+ξTtNR2−1NTξt+dtξTtMR2−1MTξt+dM−dtξTtNR2−1NTξt−ekTtΦiekt+σxTt−τtΦixt−τt,where(43)Ex˙TtPixt+xTtPix˙t=ℳTPixt+xTtPiℳ,Ex˙TtτMR1+dMR2x˙t=ℳTτMR1+dMR2ℳ+μ2NTτMR1+dMR2N,ℳ=Aixt+Biα¯Ki+ΔKixt−dt+Bi1−α¯Ki+ΔKixt−τt+ekt,N=BiKi+ΔKixt−dt+BiKi+ΔKixt−τt+ekt.
By employing Schur complement, we have the conclusion thatEV˙t<0 [48] if inequation (27) hold. Furthermore, the uncertain Markov jump system (1) is asymptotically stable. This completes the proof.
Based on the analysis and proof above, a controller will be derived and given in the next theorem for system (20).Theorem 2.
For given scalarsτM, dM, 0<σ<1 and probability parameter α¯, the closed-loop system (20) is asymptotically stable under the hybrid-driven mechanism, if there exist matrices P¯i>0, Φ¯i>0i=1,2, Q¯1>0, Q¯2>0, R¯1>0, R¯2>0, and W¯, V¯, M¯, N¯, and Yi with appropriate dimensions such that inequality (27) satisfies the following LMI:(44)Σ¯11∗∗∗Σ¯21Σ¯22∗∗Σ¯310Σ¯33∗Σ¯4100Σ¯44<0,where(45)Σ¯11=Θ¯+ϒ¯+ϒ¯T,Θ¯=Θ¯11∗∗∗∗∗Θ¯210∗∗∗∗00−Q¯1∗∗∗Θ¯4100σΦ¯i∗∗0000−Q¯2∗Θ¯610000−Φ¯i,Θ¯11=P¯iAiT+AiP¯i+λiiP¯i+Q¯1+Q¯2,Θ¯21=α¯Yi+ΔYiTBiT,Θ¯41=Θ¯61=1−α¯Yi+ΔYiTBiT,ϒ¯=M¯+W¯N¯−M¯−N¯V¯−W¯−V¯0,Σ¯21=Ξ¯1Ξ¯20Ξ¯30Ξ¯4,Σ¯31=0Ξ¯50Ξ¯60Ξ¯7,Σ¯41=τMW¯τMV¯dMM¯dMN¯,Σ¯22=Σ¯33=diagτMR¯1−2P¯i,dMR¯1−2P¯i,Σ¯44=diag−τMR¯1,−τMR¯1,−dMR¯2,−dMR¯2,Ξ¯1=τMAiP¯idMAiP¯iΞ¯2=τMα¯BiYi+ΔYidMα¯BiYi+ΔYi,Ξ¯3=Ξ¯4=τM1−α¯BiYi+ΔYidM1−α¯BiYi+ΔYi,Ξ¯5=Ξ¯6=Ξ¯7=τMμBiYi+ΔYidMμBiYi+ΔYi.Proof.
According to Theorem1, system (20) is asymptotically stable. We define the appropriate dimension matrix J=diagPi−1,Pi−1,…,Pi−1, then pre- and postmultiply (27) with J and its transposition, respectively. We define(46)Ki=YiP¯i−1,P¯i=Pi−1,W¯=P¯iWP¯i,V¯=P¯iVP¯i,M¯=P¯iMP¯i,N¯=P¯iNP¯i,R¯1=P¯iR1P¯i,R¯2=P¯iR2P¯i,Q¯1=P¯iQ1P¯i,Q¯2=P¯iQ2P¯i,Φ¯i=P¯iΦiP¯i,ΔYi=ΔKiP¯i,where Ki represents the controller gain of the closed-loop system. For simplifying calculations and derivation, inspired by Lemma 4, we can substitute −R1 and −R2 with R¯1−2P¯i and R¯2−2P¯i. According to the Schur complement, we obtain the inequality (44). This completes the proof. □Remark 4.
It is worth noting that a sufficient condition is proposed in Theorem2 referring to the co-design of the controller and hybrid-driven matrix Φi. Due to the existence of nonlinear term ΔYi in Theorem 2, it is scarcely possible for the LMI toolbox to deal with more than two unknown items. Therefore, we must tackle nonlinear term. In addition, considering the Markov jump continuous uncertain network system, we substitute Ai with Ai+DiFtHi. Then, a new technique is used to rewrite the Theorem 2 in the next section.Theorem 3.
For given scalarsτM, dM, 0<σ<1 and probability parameter α¯, the closed-loop system (20) is asymptotically stable under the hybrid-driven mechanism, if there exist matrices P¯i>0, Φ¯i>0i=1,2, Q¯1>0, Q¯2>0, R¯1>0, R¯2>0 and W¯, V¯, M¯, N¯, Yi with appropriate dimensions such that the inequality (27) satisfying the following LMI:(47)Σ^1∗Σ^2Σ^3<0,where(48)Σ^1=Σ^11∗∗∗Σ^21Σ¯22∗∗Σ^310Σ¯33∗Σ¯4100Σ¯44<0,Σ^11=Θ^+ϒ¯+ϒ¯T,Θ^=Θ¯11∗∗∗∗∗Θ^210∗∗∗∗00−Q¯1∗∗∗Θ^4100σΦ¯i∗∗0000−Q¯2∗Θ^610000−Φ¯i,Θ^21=α¯YiTBiT,Θ^41=Θ^61=1−α¯YiTBiT,Σ^21=Ξ¯1Ξ^20Ξ^30Ξ^4,Ξ^2=τMα¯BiYidMα¯BiYi,Ξ^3=Ξ^4=τM1−α¯BiYidM1−α¯BiYi,Σ^31=0Ξ^50Ξ^60Ξ^7,Ξ^5=Ξ^6=Ξ^7=τMμBiYidMμBiYi,Σ¯2=Ψ1Ψ2T,Ψ1=ν1L¯DTL¯HTκ1S¯11TS¯31T,Ψ2=κ3S¯41Tκ2S¯12TS¯32Tκ4S¯42T,Σ¯3=diagΨ3Ψ4,Ψ3=−v1I−v1I−κ1I−v2I,Ψ4=−κ3I−κ2I−ν5I−κ4I,L¯DT=DT0⋯0︸5τMDTdMDT0⋯0︸6,L¯H=HP¯i0⋯0︸13,S¯11=α¯BiT0⋯0︸5Γ10⋯0︸4,Γ1=τMα¯BiTdMα¯BiTτMμBiTdMμBiT,S¯12=1−α¯BiT0⋯0︸5Γ20⋯0︸4,Γ2=τM1−α¯BiTdM1−α¯BiTτMμBiTdMμBiT,S¯31=0Yi0⋯0︸12,S¯41=0P¯i0⋯0︸12,S¯32=0⋯0︸4Yi0Yi0⋯0︸7,S¯42=0⋯0︸4P¯i0P¯i0⋯0︸7.Proof.
On the basis of inequality (44) in Theorem 2, we can rewrite the inequation as follows:(49)Σ^1+Λ1+Λ2+Λ3<0where(50)Λ1=symLDiFtLHi,Λ2=symS11TΔgS31+symS11TKiS21+symS11TΔgKiS21,Λ3=symS12TΔgS32+symS12TKiS22+symS12TΔgKiS22.
We can obtain the following equations by applying Lemma1:(51)Λ1=ν1−1L¯DL¯DT+ν1L¯HTL¯H,Λ2=ν2−1S¯11TΔ2gS¯11+ν2S¯31TS¯31+ν3−1S¯11TS¯11+ν3S¯21TKiTKiS¯21+ν4−1S¯11TΔ2gS¯11+ν4S¯21TKiTKiS¯21,Λ3=ν5−1S¯12TΔ2gS¯12+ν5S¯32TS¯32+ν6−1S¯12TS¯12+ν6S¯22TKiTKiS¯22+ν7−1S¯12TΔ2gS¯12+ν7S¯22TKiTKiS¯22.
Synthesizing the formulas above, we get the inequation of (49):(52)Σ^1+ν1−1L¯DL¯DT+ν1L¯HTL¯H+κ1S¯11TS¯11+κ2S¯12TS¯12+ν2S¯31TS¯31+ν5S¯32TS¯32+κ3S¯41TS¯41+κ4S¯42TS¯42<0.
Noticed thatΔ2g≤δg2I and Δ2f≤δf2I, inequality (52) can be rewritten as (47) by applying Schur complement. This completes the proof.


## 4. Simulation Examples

In this section, two examples will be provided to illustrate the advantages of the controller designed in this paper.Example 1.
Consider the system parameters as follows.Mode 1:(53)A1=−0.20.60.1−1,B1=0.50.1,D1=10.5,H1=0.10.2.Mode 2:(54)A2=−10.60.6−1.5,B2=0.50.2,D2=0.10.5,H2=0.50.2.
The transition probability matrix of Markov jump process is supposed to be(55)Π=−334−4.
Then, the initial condition of the system is decided asx0=−2.52.5T. Comparing the previous investigation [35], we divide the cases into four conditions. Among the four cases, we apply the same Markov jump and the BCQ effectively while different stochastic variables are employed following the Bernoulli distribution.
Setting the corresponding parameters network-induced delayτM=0.5, dM=0.1, event-trigger scalar σ=0.05, the sampling period h=0.1s, ν1=ν2=ν4=ν5=1, and κ1=κ2=κ3=κ4=35. For simplicity, we assume that the quantization indicators of the BCQ are identical δh=δg. We obtain the following cases.Case 1.
The stochastic variableα¯=1 means that the system is working under the “time-triggered” mode. By the LMI toolbox and Theorem 3, we can obtain the controller feedback gains:(56)K1=−1.46770.7772,K2=0.3801−0.2721.
Figure2 shows the transition probability of Markov jump between different modes. Figure 3 illustrates the state response trajectories, demonstrating that the system under time-triggered scheme is asymptotically stable.Figure 2
The state transition of Markov jump in Case1.Figure 3
The state response in Case1.Case 2.
Setting the stochastic variableα¯=0, then system (20) is under the “event-triggered” communication scheme. By utilizing Theorem 3, we can obtain the controller feedback gains:(57)K1=0.5377−0.3553,K2=−0.1512−1.4329,and the event-triggered matrices(58)Φ1=9.1161−3.4920−3.49201.9255,Φ2=10.7435−4.9141−4.914111.4700.
Figures4, 5, and 6 represent the probabilities of Markov jump between different modes, the state response trajectories and the “event-triggered” release instants and interval.Figure 4
The state transition of Markov jump in Case2.Figure 5
The state response in Case2.Figure 6
The “event-triggered” release instants and interval in Case2.Case 3.
Utilizing the LMI technology of Matlab, we assume that the stochastic variableα¯=0.2 and communication mechanism is “hybrid-driven” scheme. The controller parameters subject to the following matrices:(59)K1=1.8301−0.2808,K2=−0.4947−4.0302,and the event-triggered matrices(60)Φ1=2.0808−52.8944−52.89448.9900,Φ2=17.339143.707843.70781.7041.
Figure7 illustrates the variation of state response trajectories in Case 3. Figure 8 represents the “event-triggered” release instants and interval in hybrid-driven mechanism. Figure 9 stands for the jump of Bernoulli distribution between “event-triggered” and “time-triggered.”Figure 7
The state response in Case3.Figure 8
The “event-triggered” of “hybrid-driven” release instants and interval in Case3.Figure 9
The jump of Bernoulli distribution in Case3.Case 4.
Having a resemblance to Case3, we obtain the following matrices by solving LMIs of Theorem 3 and setting stochastic variable α¯=0.6:(61)K1=0.2255−1.5036,K2=0.36042.7912,and the event-triggered matrices(62)Φ1=1.7512−17.9441−17.94410.8417,Φ2=23.514113.328913.32896.0630,
Figure10 depicts the variation of state response trajectories of Case 4. Figure 11 indicates the “event-triggered” release instants and interval in the hybrid-driven mechanism. Figure 12 stands for the jump of Bernoulli distribution between “event-triggered” and “time-triggered.”
Based on the simulation pictures, controller gains and event-triggered matrix above, we make the following detailed analysis.
As depicted in the pictures, we design the sampling periodh=0.1s and the sampling instants N=300.
During the simulation time of30s, the trigger implements only 26 times in 300 times with α¯=0.2. To compare the signal transmission rates of the system under a different Bernoulli probability within 30s, the result of the comparison is given as Table 1.
The trigger times of event-trigger are 37 withα¯=0, as shown in Figure 6. Comparatively, with the increase of value α¯, the system is under hybrid-driven mode, and the frequency of data transmission and trigger count are decreasing. When Bernoulli probability α¯=0.2, α¯=0.6, and α¯=0.9, respectively, the signals transmission rates reduce to 8.67%, 5.67%, and 1%. It is obvious that the data packet transmission rate with hybrid-driven mechanism is less than the one with event-triggered scheme. The method proposed in this paper degrades the waste of network communication resources and mitigate the burden of communication channel in control systems. It is better than the existing event-triggered mechanism investigation under the premise of asymptotically stable systems.Figure 10
The state response in Case4.Figure 11
The “event-triggered” of “hybrid-driven” release instants and interval in Case4.Figure 12
The jump of Bernoulli distribution in Case4.Table 1
Transmission rates.
α
¯
V
Transmission rates
0
37
12.33%
0.2
26
8.67%
0.6
17
5.67%
0.9
3
1%Remark 5.
Contrasting with the existing literature about the dealing method of NCSs with quantization and network-induced delay, some results with using extraordinary techniques have shown certain superiorities [8], in which two event-triggers are employed in plant-to-controller channel and controller-to-actuator channel, and the best transmission rate 12% outnumbered our simulation result. Furthermore, the system stabilization period in this paper is superior to [33].Remark 6.
In Figures7 and 10, the state response trajectories with different Bernoulli probabilities are depicted. We can easily conclude that the number of data packet transmission decreases greatly under the hybrid-driven mechanism, but the system stability time is longer than event-triggered scheme. It proves the standpoint in literature [35] that we should get a balance between the system performance and resource saving.Example 2.
In this section, a system of single-link robot arm is considered [16, 49] and given by(63)x˙1t=x2t,x˙2t=−glMJsinx1t−RJx2t+1Jut,where x1t is the angle of the robot arm from the vertical and x2t denotes the angular velocity. The system parameters M, J, R, g, and l represent the mass of the load, the moment of inertia, the damping coefficient, the gravity constant, and the length of the arm, respectively.
Being similar to Example1, the sampling period is set as T=0.1s and the sampling instants N=300. The stochastic variable is given as α¯=0.5 to illustrate that the hybrid-driven mechanism with BCQ for the Markov jump system is stable when it is introduced in the real system. It is worth mentioning that the real system parameters Mi and Ji have two different modes, which are displayed in Table 2, and the transition probability matrix is maintained.
Then, the real system of the single-link robot arm can be presented as following based on hybrid-driven mechanism for the Markov jump system:(64)x˙t=Ai+ΔAixt+Biut,where(65)Ai=−1T−TglMiJi1−TRJi,Bi=−0TJi,ΔAi=DiFtHi,Di=01,Hi=00.1,with the calculation of Theorem 3 by LMT toolbox technology, and the real system controller gains are(66)K1=0.0011−0.0231,K2=0.0010−0.0212,and the corresponding event-triggered matrices(67)Φ1=54.6512−1.4947−1.494714.0670,Φ2=33.40384.10974.109717.3571.
In this example, Figure13 expresses the Markov stochastic jump for transition probability. It is apparent from Figure 14 that the real system of the single-link robot arm is asymptotically stable under the approach proposed in this paper. Figure 15 denotes the “event-triggered” release instants and interval in hybrid-driven mechanism of Example 2. As shown in Figures 13–15, the hybrid-driven mechanism with BCQ for the Markov jump system can be employed into the real system and the stability is guaranteed.Table 2
Different modes ofMi and Ji.
Mode
i
=
1
i
=
2
M
i
1
4
J
i
1
4Figure 13
The Markov jump of Example2.Figure 14
The state response of Example2.Figure 15
The “event-triggered” of “hybrid-driven” release instants and interval of Example2.


## 5. Conclusion

The hybrid-driven mechanism problem of controller design for the Markov jump system with both channel quantization and network-induced delay based on uncertain networked is developed in this paper. The hybrid-driven mechanism is employed for the balance between data transmission rate and system performance. Both channel quantization technology is applied to reduce network communication resource utilization further. The numerical simulations and the practice example illustrate the effectiveness of the proposed method in this paper.Nevertheless, with the development of information science and communication technology, the network security issue is attracting extensive attention and interest. How to defend the cyber attacks is the present research focus. Thus, the approach of unifying the different varieties of cyber attacks, for instance, denial of service (DoS) attacks and deception attacks into a Markov jump system remains to further be discussed.



---

*Source: 3148252-2020-04-08.xml*