# Event-Triggered Stability Analysis of Semi-Markovian Jump Networked Control System with Actuator Faults and Time-Varying Delay via Bessel–Legendre Inequalities

Authors: Hongqian Lu; Chaoqun Guo; Yue Hu; Wuneng Zhou
Journal: Complexity
 (2019)

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## Abstract

This paper discusses the stability of semi-Markovian jump networked control system containing time-varying delay and actuator faults. The system dynamic is optimized while the network resource is saved by introducing an improved static event-triggered mechanism. For deriving a less conservative stability criterion, the Bessel–Legendre inequalities approach is employed to the stability analysis and plays a major role. By constructing the enhanced Lyapunov–Krasovskii functional (LKF) relevant to the Legendre polynomials, a stability criterion with lower conservativeness indexed byN is derived, and the conservativeness will decrease as N increases. In addition, a controller is designed. To prove the validity of this paper, numerical examples are provided at the last.


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## Body

## 1. Introduction

Networked control system is widely used and researched due to its many advantages such as sharing network resources, cutting down system cost, and improving system stability. The networked control system is a feedback control system consisting of the controller, actuator, controlled object, and sensor [1]. There are many unavoidable problems in the networked control system. Among them, the network time delay has a large impact on stability of the networked control system [2, 3]. The networked control time delay occurs in the transmission process of information. This paper ignores the process delay caused by the calculation of sensors, controllers, and actuators themselves and only considers the network time delay which is called the network time-varying delay. In addition, an actual networked control system may suffer from actuator faults resulting in losing partial or total effectiveness of executing control actions, which can also cause instability of the networked control system. Thus, researching stability of the networked control system containing actuator faults is of vital importance [4]. Observer-based fault estimation for a nonlinear system was studied in [5]. A reliable controller for system with actuator faults was researched in [6]. Chance constrained control was proposed in [7] for stochastic nonlinear systems.As a result of the inevitable external interferences, parameters and structures of the model may suddenly change. The jumping of parameters is usually subject to Markovian jump process. Thus, the Markovian jump has received wide attention in control systems. It is a useful tool in constructing the model structures which could abrupt change in their parameters [8]. Many research results about the Markovian jump systems are proposed. The stability of the Markovian jump networked control system was analyzed in [9] and the filter problem of the networked control system using the Markovian jump method was researched in [10]. It is worth mentioning that the actuator faults usually have the random property like Markovian jump. Literatures [11–13] all studied the control problem of the system with Markovian jump actuator faults. However, the application of the Markovian jump structure is limited because of the assumption that sojourn-time l obeys exponential distribution, and the transition rate is a constant. Nevertheless, in the practical applications, such an assumption is not always satisfied. To overcome this defect, we slack probability distribution of sojourn-time as a general probability distribution. This will make the transition rate become time-varying [14–16]. Correspondingly, this stochastic process is often called as semi-Markovian jump process. Nowadays, there are many research results about semi-Markovian jump. For instance, robust sliding mode control of the uncertain semi-Markovian jump system was researched in [17]. An observer was established for the fuzzy system containing stochastic actuator faults in [18]. This paper will further study the stability of the semi-Markovian jump networked control system with actuator faults and delays.For cutting down the conservatism of the delay upper bound, in terms of technology, a series of approaches are developed aiming to decrease the approximation error of term∫t−d2tx˙TsRx˙sds in the derivative of LKF. At first, a model transformation method is employed to deal with that integral term, which has large conservatism [19]. Afterwards, a free weighting matrix method was studied in [20] to reduce the conservatism of stability criterion. Nevertheless, this approach will raise decision variables. Then, Jensen inequality was widely employed to overcome this defect [21]. Besides, Wirtinger-based inequality, as a more advanced technology than Jensen inequality, was introduced in various control systems [22]. Moreover, free-matrix-based integral inequality was applied in [23] for researching filtering problem for the neural network system. A kind of integral inequality method based on auxiliary function was introduced in [24]. These techniques are used to approximate the derivative of the LKF more accurately so as to obtain less conservative linear matrix inequalities in the stability criterion. But, they all have some manipulations permitting to use a finite linear matrix inequality to test the infinite dimensional stability problem [25–28]. In this paper, we use the Bessel–Legendre inequalities method to tightly approximate the integral term of quadratic functional to the quadratic term of matrix functional about Legendre polynomials. For example, term “−∫t−τMtx˙tWx˙tds” in Lyapunov–Krasovskii functional derivative is less than or equal to the “−τMΨNTWNΨN”, where WN=diagW,3W,…,2N+1W, ΨN=1/τM∫t−τMtℒNs−τM/τMxsds, N≥0, and the ℒN is the “shifted” Legendre polynomial matrix. Moreover, we construct an improved LKF which is relevant to Legendre polynomials and make use of the reciprocally convex combination method [29–31] to obtain the stability criterion. The derived stability criterion with lower conservativeness is governed by the order N. The conservatism of the criterion decreases as N increases. To the author’s best knowledge, for the semi-Markovian jump networked control system containing actuator faults and time-varying delay, employing Bessel–Legendre inequalities to investigate stability has not been fully researched and is of vital significance.In fact, network channel resources are limited. In the past, most articles applied a periodic triggered mechanism to study networked control systems. The traditional periodic triggered mechanism is fixed to deal with various disturbances. This will cause an increase in the amount of computation of controller because the system is not always under disturbance [32–34]. The event-triggered scheme can decrease the waste of channel resources effectively compared with the traditional periodic triggering scheme. Thus, event-triggered stability analysis not only guarantees the stability of system but also reduces the burden of channel. The distribute control under the event-triggered scheme was researched in [35]. An event-triggered controller for the discrete networked control systems was designed in [36]. The event-triggered real-time scheduling strategy was introduced in [37] for the T-S fuzzy system. Recently, the researchers have developed some improved ETSs for different control systems and different property requirements. Decentralized ETS was researched for the networked fuzzy system in [38]. Distributed ETS for a multiagent system was dealt in [39]. In terms of accelerating system dynamics, we usually expect the high frequency of release at the beginning process. A new static ETS was introduced for a discrete nonlinear system in [40]. Correspondingly, the dynamic ETS which contains the dynamic variables was studied for the networked multiagent system in [41]. Now, in this paper, for the purpose of improving system dynamics while reducing transmission burden, we will introduce an improved static ETS for this semi-Markovian jump networked control system containing actuator faults and time-varying delay.Thus, there exist several questions to further investigate the semi-Markovian jump networked control system containing actuator faults and delay based on the Bessel–Legendre inequalities approach as follows:(1)
Is it feasible to complete the analysis of this semi-Markovian jump networked control system containing actuator faults and network delay by employing the Bessel–Legendre inequalities approach?(2)
How to establish an improved ETS for this comprehensive continuous system with time delay to raise the release frequency in beginning stage and then gradually reduce the release rate? Such an ETS can not only reduce the waste of channel resources but also accelerate the system dynamics.(3)
Based on the approach of Bessel–Legendre inequalities, how can we design an effective event-triggered controller to control the semi-Markovian jump networked control system containing actuator faults and time-varying delay?Despite all questions mentioned above are necessary to solve, no one has ever carried out. Now, this paper is committed to dealing with these problems.In general, the major contributions are (1) A comprehensive model such as the semi-Markovian jump networked control system with actuator faults and time-varying delay is researched. Different from previous research studies, this paper applies Bessel–Legendre inequalities approach to analyze the stability of the system and constructs an appropriate Lyapunov–Krasovskii functional with respect to Legendre polynomials. Finally, an affine linear matrix inequality is obtained. The acquired stability criterion which is indexed byN has lower conservatism. And the larger N is, the lower conservatism is. (2) This paper establishes an improved static ETS for this comprehensive continuous system to effectively shorten system dynamic process and decrease the burden of transmission. (3) A valid event-triggered controller is ingeniously designed.The remainder of this paper can be summarized as below. Section2 shows definitions and problem formulation. In addition, an improved static ETS is established. In Section 3, the result of stability analysis is presented. An effective controller is designed in Section 4. Section 5 provides the numerical examples to verify the effectiveness of the research results. Finally, Section 6 shows the conclusions.Notations.
ℰ
⋅ denotes the mathematical expectation. ℕ denotes the natural number. S+N is the set of symmetric positive definite matrices of ℝn×n, and ℝn×n represents the set of all n×n matrices. For any matrix A, HeA=A+AT.


## 2. Definitions and Problem Formulation

The general state space model of the semi-Markovian jump networked control system can be expressed as follows:(1)x˙t=Artxt+Brtut,where xt∈ℝn is the system state and ut∈ℝp is the control input. When t≥0, rt is the semi-Markovian jump process and rt=i∈S, S=1,2,…,M. At t=0, the initial value x0=x0, and the initial mode of the semi-Markovian jump process is r0=r0. Art is system matrix with appropriate dimensions. Due to rt=i, then Art is expressed by Ai.The sojourn time of the semi-Markovian jump process does not follow the exponential distribution. Thus, the semi-Markovian jump process has the following transition probabilities:(2)Prrt+l=jrt=i=λijll+ol,i≠j,1+λiill+ol,i=j,and l is the sojourn time. λijl denotes the transition rate from mode i at time t to mode j at time t+l for i≠j. λiil=−∑j=1,j≠iMλijl. ol is the high order infinitesimal of l and liml⟶0ol/l=0. We define that λil is the system transition rate of mode i, and then the λil satisfies(3)λil=fil1−Fil.At modei, function fil denotes the probability distribution of sojourn time and function Fil denotes cumulative distribution of sojourn time.Due to that the bandwidth is limited and the capability of the controller, sensor, and actuator is finite, the traditional periodic-triggered mechanism which will send some unnecessary signals under the most cases without disturbances can cause waste of network resources. In order to shorten the system dynamics while decreasing the load of network transmission, this paper introduces an improved event-triggered scheme.Assume that the sampled states att0h,t1h,t2h,…, satisfy event-triggering condition, then we call the t0h,t1h,t2h,…, as release times. The signals at t0h,t1h,t2h,…, can be sent from the sensor to controller. However, there exists a time delay when signal arrives at the actuator from controller. Suppose that τk∈0,τ¯ is the network time delay and τ¯ is a positive scalar. Thus, xt0h,xt1h,xt2h,… are transmitted to the actuator at the time t0h+τ0,t1h+τ1,t2h+τ2,…, respectively. We redescribe system (1) as(4)x˙t=Aixt+Biutkh.Due tout=Krtxt, thus,(5)x˙t=Aixt+BiKixtkh.Considering the time delay in the network transmission process, we need to set up the model of the network time delay. Define that(6)ϕk=minαtkh+τk+αh≥tk+1h+τk+1,α=0,1,2,….Let(7)Iα=tkh+τk+α−1h,tkh+τk+αh,Iϕk=tkh+τk+ϕk−1h,tk+1h+τk+1,α=1,2,…,ϕk−1.Then,(8)tkh+τk,tk+1h+τk+1=∪α=1ϕkIα.Fort∈tkh+τk,tk+1h+τk+1, define that(9)τt=t−tkh,t∈I1,t−tkh−h,t∈I2,⋅⋅⋅⋅⋅⋅t−tkh−ϕk−1h,t∈Iϕk,ekt=0,t∈I1,xtkh−xtkh+h,t∈I2,⋅⋅⋅⋅⋅⋅xtkh−xtkh+ϕk−1h,t∈Iϕk.Apparently,0<τk≤τt≤τ¯+h. Let τM=τ¯+h, then 0<τk≤τt≤τM.We expect the proposed ETS to achieve the situation that more packets are transmitted during transient response to make the system approach stability faster. And when the system nears the steady state, the triggering should be limited. Next, we introduce the following improved static event-triggered mechanism:(10)ekTtΛekt≤σktxTt−τtΛxt−τt,where Λ is symmetric positive definite matrix, and time-varying function σkt meets(11)σkt=σkα,t∈Iα,α=1,2,…,ϕk,σkα+1=σ¯+ekαTekαε+ekαTekασkα−σ¯,ekα=xtkh−xtkh+α−1h,α=1,2,…,ϕk,where σ¯ is the upper bound of σkα and σ¯∈0,1, 0<σk1<σ¯, known constant ε>0.Usingekt and τt, for t∈tkh+τk,tk+1h+τk+1, we rewrite (5) as(12)x˙t=Aixt+BiKixt−τt+BiKiekt.In practical systems, the actuators may experience failures, which may cause the instability of real system. Therefore, considering the actuator faults is meaningful, under the even-triggered mechanism, we defineuftkh as the control signal sent from the actuator and(13)uftkh=βrtutkh,where βrt=diagβ1rt,β2rt,…,βprt and 0≤β¯wrt≤βwrt≤β¯wrt≤1, w=1,2,…,p. The β¯wrt and β¯wrt denote the failure bounds of the pth actuator under the fault mode i. We use βi denoting βrt, then βi=diagβ1i,β2i,…,βpi with 0≤β¯wi≤βwi≤β¯wi≤1. It is easy to see that if β¯wi=β¯wi=0, then the w th actuator completely outage in the ith fault mode; if β¯wi=β¯wi=1, then the w th actuator under fault mode i has no failure. Define(14)β^i=diagβ¯1i+β¯1i2,β¯2i+β¯2i2,…,β¯pi+β¯pi2,βˇ=diagβ¯1i−β¯1i2,β¯2i−β¯2i2,…,β¯pi−β¯pi2.Rewriteβi as(15)βi=β^i+Δi,where Δi=diagδ1i,δ2i,…,δpi, and δwi≤β¯wi−β¯wi/2, w=1,2,…,p.Next, the complete model is shown as(16)x˙t=Aixt+BiβiKixt−τt+BiβiKiekt,xt=ϕt,t∈−τM,0,where the function Φt is continuous on −τM,0. Note that(17)0<τk≤τt≤τM,d1≤τ˙t≤d2,where scalars d1<0 and d2>0.This paper analyzes the stability of system (16) via the Bessel–Legendre inequalities. Before the stability research, we first give some definitions and lemmas to facilitate analysis.Lemma 1 (see [42]).
Assume thatW1,W2∈ℝn are symmetric positive matrices. If there exist the symmetric matrices X1,X2∈ℝn and the matrices Y1,Y2∈ℝn such that for ε=0,1, the(18)W100W2−εX1Y1Y1T0−1−ε0Y2Y2T0≥0,holds, then the following inequality is true for all ε∈0,1:(19)1εW10011−εW2≥W100W2+ε0Y1Y1TX2+1−εX1Y2Y2T0.Next, we give the definition of the “shifted” Legendre polynomials which is defined over the interval0,1.Definition 1.
For allu∈0,1, f,q∈ℕ, the “shifted” Legendre polynomials is the following formula:(20)Lfu=−1f∑q=0fρqfuq,where ρqf=−1qfqf+qq, fq means the binomial coefficients, and fq=f!/f−q!q!.
According to the Legendre polynomials, we define polynomial matrixLN:(21)LNu≔L0uInL1uIn…LNuInT,where n∈ℕ and N∈ℕ. The Legendre polynomials have the orthogonality property which results in the application of the Legendre polynomials. For any symmetric positive definite matrix W(22)∫01LNuW−1LNTudu=W¯N−1,is true, where W¯N=diagW,3W,…,2N+1W.
In the process of stability analysis, we need to use the following techniques with respect to the differential about the Legendre polynomials matrix:(23)dduLNu=ϒ¯NLNu=ϒNLN−1u,(24)dduuLNu=LNu+ΞNLNu,where ϒ¯N=ϒN0nN+1,n, ϒN=υN⊗In, and ΞN=e^N⊗In. The matrix υN∈ℝN+1×N is defined as(25)υNf,q=0,iff≥q,2f−11−−1f+q,iff<q.
The matrixe^N∈ℝN+1×N+1 is defined as(26)e^Nf,q=0,iff>q,f,iff=q,2f−1,ifk<q.
Noting that the “shifted” Legendre polynomials are defined on the interval0,1, we give the evaluation values of the polynomial matrix at u=0 and u=1:(27)LN0=In−In⋮−1NIn≔χ¯N,(28)LN1=InIn⋮In≔χN.
The next lemma expresses the Bessel–Legendre inequality.Lemma 2 (see [43]).
Forx∈ℒ2a,b⟶ℝn, any appropriate dimensional matrix W which is a symmetric positive definite, and any N∈ℕ, the inequality(29)∫abxTsWxsds≥b−aΨNTW¯NΨN,holds, where(30)ΨN=1b−a∫abLNs−ab−axsds,W¯N=diagW,3W,…,2N+1W.Definition 2.
For anyr0∈S and any initial state Φt∈−τM,0, if the following inequality(31)limt⟶∞ℰ∫0txs2dsΦt,r0≤∞,is true, then system (16) is stochastically stable.


## 3. Stability Analysis

In this part, we research the stochastic stability of system (16). The less conservative stability criterion is given in Theorem 1.Theorem 1.
GivenN∈ℕ, scalar ε>0, if there exist matrix PNj∈S+2N+3n, matrices Q1,Q2,W∈S+n, and matrices Y1,Y2∈ℝN+1n×N+1n such that for all τt,τ˙t∈ℋ the following inequality(32)ΘN′=Θ¯NΠNTτM−τtτMY10ΠNT0τtτMY2TτMHNT∗−τM−τtτMW¯N00∗∗−τtτMW¯N0∗∗∗W−2In≤0,holds, then system (16) is stochastically stable, where(33)ℋ=Co0,0,0,d2,τM,0,τM,d1,Θ¯N=HeE1,N+τ˙tE2,NTPNiΦN+ΦNT∑j=1Mλ¯i,jPNjΦN+E3,N−∏NTΛNΠN,ΠN=χN−ϒ¯N0−χ¯N0000−ϒ¯NχN−χ¯N0,W¯N=diagW,3W,…,2N+1W,HN=Ai00BiβiKi0BiβiKi,E1,N=Ai00BiβiKi0BiβiKiχN−ϒ¯N0−χ¯N0000−ϒ¯NχN−χ¯N0,E2,N=0000000−ΞN+ϒ¯N0χ¯N0000ΞN−χ¯N00,ΦN=In000000τtInN000000τM−τtInN000,E3,N=diagQ1,0,0,1−τ˙tQ2−Q1+σktΛ,−Q2,−Λ,ΛN=W¯N00W¯N+τtτM0Y1Y1TW¯N+τM−τtτMW¯NY2Y2T0,and the unit matrix InN∈ℝN+1n×N+1n and In∈ℝn×n.Proof.
Firstly, we construct the LKF as(34)VNxt,x˙t,rt,t=V1Nxt,rt,t+V2Nxt,t+V3Nxt,x˙t,t,where(35)V1Nxt,rt,t=ξNTtPNrtξNt,V2Nxt,t=∫t−τttxTsQ1xsds+∫t−τMt−τtxTsQ2xsds,V3Nxt,x˙t,t=τM∫t−τMt∫θtx˙TsWx˙sdsdθ,where(36)ξNTt=xTtτtΨ1,NTtτM−τtΨ2,NTt,Ψ1,Nt=1τt∫t−τttLNs−t+τtτtxsds,Ψ2,Nt=1τM−τt∫t−τMt−τtLNs−t+τMτM−τtxsds.
For deriving the stability criterion, we fist calculate the weak infinitesimal generatorℒVN of LKF. The main task is to obtain an upper bound of the weak infinitesimal generator ℒVN by employing the Bessel–Legendre polynomials method. Through using Lemma 1 and Lemma 2, we will carry out a series of relaxations and simplifications on ℒVN to get a tight upper bound. Next, we calculate the ℒVN:(37)ℒVN=ℒV1N+∑k=23ℒVkN.
Among them,(38)ℒV1N=limΔ⟶0ℰV1Nt+Δ,rt+Δxt,rt−V1Nxt,rtΔ.
Due tort=i∈S, making use of transition probability of the semi-Markovian jump process, we have(39)ℒV1N=limΔ⟶01Δℰ∑j=1,j≠iMPrrt+Δ=jrt=iξNTt+ΔPNjξNt+Δ+Prrt+Δ=irt=iξNTt+ΔPNiξNt+Δ−ξNTtPNiξNt=limΔ⟶01Δℰ∑j=1,j≠iMqijFil+Δ−Fil1−FilξNTt+ΔPNjξNt+Δ+1−Fil+Δ1−FilξNTt+ΔPNiξNt+Δ−ξNTtPNiξNt.
The derivation process of equation (39) is aided by (3), and the following definitions that λijl=△qijλil,i≠j, and λiil=△−∑j=1,j≠iMλijl, where qij is the transition probability intensity, and qii=−∑j=1,j≠iMqij=−1 [44]. In equation (39), Prrt+Δ=jrt=i=λijlΔ=qijλilΔ=qijfilΔ/1−Fil=qijFil+Δ−FilΔ/1−FilΔ=qijFil+Δ−Fil/1−Fil,Prrt+Δ=irt=i=1+λiiΔ=1−∑j=1,j≠iMλijlΔ=1−∑j=1,j≠iMqijFil+Δ−Fil/1−Fil=1−Fil+Δ/1−Fil.
WhenΔ is small, we can get that(40)ξNt+Δ=Δξ˙Nt+ξNt+oΔ.
Then, (39) becomes(41)ℒV1N=limΔ⟶01Δℰ∑j=1,j≠iMqijFil+Δ−Fil1−FilΔξ˙Nt+ξNt+oΔTPNj⋅Δξ˙Nt+ξNt+oΔ+1−Fil+Δ1−FilΔξ˙Nt+ξNt+oΔT⋅PNiΔξ˙Nt+ξNt+oΔ−ξNtTPNiξNt.
Due to(42)limΔ⟶0Δ2Δ=0,limΔ⟶0qijFil+Δ−Fil1−Fil=0,then(43)ℒV1N=limΔ⟶01Δℰ∑j=1,j≠iMqijFil+Δ−Fil1−FilξNTtPNjξNt+ℰ1−Fil+Δ1−Filξ˙NTtPNiξNt+ξNTtPNiξ˙Nt+1Δ1−Fil+Δ1−FilξNTtPNiξNt−1ΔξNTtPNiξNt,=limΔ⟶01Δℰ∑j=1,j≠iMqijFil+Δ−Fil1−FilξNTtPNjξNt+ℰ1−Fil+Δ1−Filξ˙NTtPNiξNt+ξNTtPNiξ˙Nt+1ΔFil−Fil+Δ1−FilξNTtPNiξNt.
According to(44)limΔ⟶01−Fil+Δ1−Fil=1,we can get(45)ℒV1N=ℰξ˙NTtPNiξNt+ξNTtPNiξ˙Nt+limΔ⟶01ΔFil+Δ−Fil1−Fil∑j=1,j≠iMqijξNTtPNjξNt−ξNTtPNiξNt.
Notice that(46)limΔ⟶01ΔFil+Δ−Fil1−Fil=11−FillimΔ⟶0Fil+Δ−FilΔ,according to (3) and (46) becomes λil. Then, ℒV1N becomes (47)ℒV1N=ℰξ˙NTtPNiξNt+ξNTtPNiξ˙Nt+ξNTt∑j=1MλijlPNjξNt.
By the same approach in [44],(48)ℒV1N=ξ˙NTtPNiξNt+ξNTtPNiξ˙Nt+ξNTt∑j=1Mλ¯ijPNjξNt,where λ¯ij≔ℰλijl=∫0∞λijlfildl.
Before solvingℒV2N and ℒV3N, we do some processes on (48) in order to conveniently derive inequality (32) in Theorem 1. We need to define that(49)ηNTt=xTtΨ1,NTtΨ2,NTtxTt−τtxTt−τMekTt.Then,(50)ξNTt=ηNTtΦNT,where ΦN is defined in Theorem 1. In addition,(51)ξ˙NTt=x˙TtddtτtΨ1,NTtddtτM−τtΨ2,NTt.
Among them,(52)x˙t=Aixt+BiβiKixt−τt+BiβiKiekt,Ψ1,Nt=1τt∫t−τttLNs−t+τtτtxsds,Ψ2,Nt=1τM−τt∫t−τMt−τtLNs−t+τMτM−τtxsds.
Setλ=s−t+τt/τt, then sλ=λτt+t−τt, and rewrite Ψ1,Nt as(53)Ψ1,Nt=∫01LNλxsλdλ.Then,(54)ddtτtΨ1,Nt=τ˙tΨ1,Nt+τt∫01LNλx˙sλλτ˙t+1−τ˙tdλ=τ˙tΨ1,Nt+τ˙tτt∫01λLNλx˙sλdλ+1−τ˙tτt∫01LNλx˙sλdλ.
Using the subsection integration method and employing (24) and (28), we can get(55)τt∫01λLNλx˙sλdλ=∫01λLNλdxsλ=LN1xt−∫01xsλddλλLNλdλ=LN1xt−Ψ1,Nt−ΞNΨ1,Nt=χNxt−Ψ1,Nt−ΞNΨ1,Nt.
In addition, we deal withτt∫01LNλx˙sλdλ. Similarly, using the subsection integration and employing (23) and (27), we obtain(56)τt∫01LNλx˙sλdλ=∫01LNλdxsλ=LN1xt−LN0xt−τt−∫01xsλddλLNλdλ=χNxt−χ¯Nxt−τt−ϒ¯NΨ1,Nt.
Thus,(57)ddtτtΨ1,Nt=τ˙tΨ1,Nt+τ˙tχNxt−Ψ1,Nt−ΞNΨ1?Nt+1−τ˙tχNxt−χ¯Nxt−τt−ϒ¯Ψ1,Nt.
Next, we deal withd/dtτM−τtΨ2,Nt. Set u=s−t+τM/τM−τt and su=uτM−uτt+t−τM, and then we rewrite Ψ2,Nt as(58)Ψ2,Nt=∫01LNuxsudu.
Then,(59)ddtτM−τtΨ2,Nt=−τ˙tΨ2,Nt+τM−τt∫01LNux˙su−uτ˙t+1du=−τ˙tΨ2,Nt−τ˙tτM−τt∫01uLNux˙sudu+τM−τt∫01LNux˙sudu,where(60)τM−τt∫01uLNux˙sudu=∫01uLNudxsu=LN1xt−τt−Ψ2,Nt−ΞNΨ2,Nt=χNxt−τt−Ψ2,Nt−ΞNΨ2,Nt,τM−τt∫01LNux˙sudu=∫01LNudxsu=LN1xt−τt−LN0xt−τM−ϒ¯NΨ2,Nt=χNxt−τt−χ¯Nxt−τM−ϒ¯NΨ2,Nt.
Thus,d/dtτM−τtΨ2,Nt becomes(61)ddtτM−τtΨ2,Nt=−τ˙tΨ2,Nt−τ˙tχNxt−τt−Ψ2,Nt−ΞNΨ2,Nt+χNxt−τt−χ¯Nxt−τM−ϒ¯NΨ2,Nt.
Combiningx˙t and (57) and (61), we can obtain(62)ξ˙Nt=E1,N+τ˙tE2,NηNt,where E1,N and E2,N are defined in Theorem 1.
Now, we use the obtained (50) and (62) to rewrite the ℒV1N in (48),(63)ℒV1N=ηNTtHeE1,N+τ˙tE2,NTPNiΦNηNt+ηNTtΦNT∑j=1Mλ¯i,jPNjΦNηNt.
Next, we will giveℒV2N and ℒV3N. Add term ekTtΛekt−ekTtΛekt on ℒV2N, then(64)ℒV2N=xTtQ1xt+1−τ˙txTt−τtQ2−Q1xt−τt−xTt−τMQ2xt−τM+ekTtΛekt−ekTtΛekt≤ηNTtE3,NηNt,where the matrix E3,N is defined in Theorem 1.(65)ℒV3N=τM2x˙TtWx˙t−τM∫t−τMtx˙TsWx˙sds=τM2ηNTtHNTWHNηNt−τM∫t−τMtx˙TsWx˙sds,where(66)HN=Ai00BiβiKi0BiβiKi.
Based on Lemma 2, we have(67)τM∫t−τMtx˙TsWx˙sds≥Ψ¯1,NΨ¯2,NTτMτtW¯N00τMτM−τtW¯NΨ¯1,NΨ¯2,N,where(68)Ψ¯1,N=1τt∫t−τttLNs−t+τtτtx˙sds,Ψ¯2,N=1τM−τt∫t−τMt−τtLNs−t+τMτM−τtx˙sds.
Employing the subsection integration method, we derive(69)τtΨ¯1,NτM−τtΨ¯2,N=χNxt−χ¯Nxt−τt−ϒ¯NΨ1,NχNxt−τt−χ¯Nxt−τM−ϒ¯NΨ2,N=ΠNηNt,where(70)ΠN=χN−ϒ¯N0−χ¯N0000ϒ¯NχN−χ¯N0.
Thus,(71)−τM∫t−τMtx˙TsWx˙sds≤−ηNTt∏NTτMτtW¯N00τMτM−τtW¯NΠNηNt.
Next, using Lemma 1 and 2 and choosingX1=WN−Y1WN−1Y1T, X2=WN−Y2TWN−1Y2 [43], we get the following inequality:(72)−τM∫t−τMtx˙TsWx˙sds≤−ηNTt∏NTΛNΠNηNt+ηNTt∏NTΛN′ΠNηNt,where(73)ΛN=W¯N00W¯N+τtτM0Y1Y1TW¯N+τM−τtτMW¯NY2Y2T0,ΛN′=τM−τtτMY1W¯N−1Y1T00τtτMY2TW¯N−1Y2.
Then, we rewriteℒV3N as follows:(74)ℒV3N=ηNTtτM2HNTWHN+∏NTΛN′ΠN−∏NTΛNΠNηNt.
Combining (63), (64), and (74), ℒVN satisfies that(75)ℒVN≤ηNTtHeE1,N+τ˙tE2,NTPNiΦN+ΦNT∑j=1Mλ¯i,jPNjΦN+E3,N+τM2HNTWHN+∏NTΛN′ΠN−∏NTΛNΠNηNt=ηNTtΘNηNt,where(76)ΘN=Θ¯N+∏NTΛN′ΠN+τM2HNTWHN,Θ¯N=HeE1,N+τ˙tE2,NTPNiΦN+ΦNT∑j=1Mλ¯i,jPNjΦN+E3,N−∏NTΛNΠN.
According to(77)In−WW−1In−W>0,thus(78)−W−1≤−2In−W.
By Schur complement, ifΘN≤0, then ΘN′≤0 which is defined in (32) in Theorem 1. As we can see, ΘN′ is multiaffine on τt and τ˙t, where τt,τ˙t∈ℋ=Co0,0,0,d2,τM,0,τM,d1. There exists ε>0 such that ℒVN≤−εxt2. Thus, system (16) is stochastically stable for any delay satisfying the set ℋ. The proof is complete.
It is should be mentioned that the solution of LMI in Theorem 1 with allowable delay setℋ has lower conservatism than that with allowable delay set 0,τM×d1,d2. In fact, in the instance of set 0,τM×d1,d2, the vertices 0,d1 and τM,d2 cannot be reached at any time for the impossible circumstances that τ˙t is negative when τt=0 and τ˙t is positive when τt=τM.


## 4. Stabilization Analysis

This section studies the stabilization for system (16). We will design a controller based on Theorem 1. The next theorem shows the main result.Theorem 2.
GivenN∈ℕ, scalar ε>0, if there exist P^1,Ni∈S+n, P^2,Ni∈S+N+1n, P^3,Ni∈S+N+1n, matrices Q1,Q2,W∈S+n, and matrices Y1,Y2∈ℝN+1n×N+1n such that the following inequality(79)Θ^N′=Θ¯N′∏NTτM−τtτMY10∏NT0τtτMY2TτMH^NT∗−τM−τtτMW¯N00∗∗−τtτMW¯N0∗∗∗W−2I≤0,is true for any τt,τ˙t∈ℋ, then system (16) is stochastically stable, where(80)ℋ=Co0,0,0,d2,τM,0,τM,d1,Θ¯N′=HeE1,N′+τ˙tE2,NTPNiΦN+ΦNT∑j=1Mλ¯i,jPNjΦN+E3,N−∏NTΛNΠN,E1,N′=Ai00P^1,Ni−2In0P^1,Ni−2InχN−ϒ¯N0−χ¯N0000−ϒ¯NχN−χ¯N0,PNi=P^1,Ni000P^2,Ni000P^3,Ni,H^N=Ai00P^1,Ni−2In0P^1,Ni−2In,and other terms are defined in (33).Proof.
If (33) holds, system (16) is called stochastically stable. In Theorem 1, there exist some nonlinear terms like E1,NTPNiΦN because of the existence of Ki. Thus, we need to eliminate the nonlinear terms in inequality (32).
Due to theBiβiKi∈ℝn×n is just one term in E1,N, and PNi is in ℕ2N+3n×2N+3n, thus we divide matrix PNi into blocks. Then, the PNi becomes that(81)PNi=P^1,Ni000P^2,Ni000P^3,Ni,where P^1,Ni∈S+n, P^2,Ni∈S+N+1n, and P^3,Ni∈S+N+1n.
Obviously,KiTβiTBiTP^1,Ni and P^1,NiBiβiKi are nonlinear terms in E1,NPNi. To eliminate the nonlinear terms, we define Ki=−βi−1Bi−1P^1,N−1i. Then, the KiTβiTBiTP^1,Ni and P^1,NiBiβiKi become −In.
In addition, the termBiβiKi which is in HN in (32) will become −P^1,N−1i. Due to(82)In−P^1,NiP^1,N−1iIn−P^1,Ni≥0,thus(83)−P^1,N−1i≤P^1,Ni−2In.
We replaceBiβiKi in (32) with P^1,Ni−2In, then we obtain(84)H^N=Ai00P^1,Ni−2In0P^1,Ni−2In.
Thus, we get(85)Θ^N=Θ¯N′+∏NTΛN′ΠN+τM2H^NTWH^N,where(86)Θ¯N′=HeE1,N′+τ˙tE2,NTPNiΦN+ΦNT∑j=1Mλ¯i,jPNjΦN+E3,N−∏NTΛNΠN.
Π
N and ΛN are defined in (33) and PNi is defined in (81). By Schur complement, if Θ^N≤0, then Θ^N′≤0 which is defined in Theorem 2. Obviously, Θ^N′≥ΘN′ which is defined in Theorem 1, thus if Θ^N′≤0, then ΘN′≤0. According to Theorem 1, system (16) is stochastically stable for any delay satisfying ℋ. The proof is complete.


## 5. Numerical Examples

Example 1.
Consider the following event-triggered semi-Markovian jump networked control system with actuator faults and time-varying delay:(87)A1=−2.11.61.5−1.9,A2=−21.7−1.2−2.2,B1=11,B2=0.90.9,λ¯i,j=0.8−0.8−0.50.5.
Takeσ¯=0.3, σk1=0.01, ε=0.01, h=0.18, β1=0.3, β2=0.2, N=0. The designed controller gains K1 and K2 and triggering parameters Λ1 and Λ2 at different times are given in Table 1. The state trajectories xt and semi-Markovian chain rt of the system are shown in Figure 1.
Under the improved static ETS, the time-varying parameterσkt and the situation of release are shown in Figure 2. From Figure 2, σkt changes from 0.01 to 0.3 and the release frequency at the initial stage is larger than other times. In addition, only 25.2% sampled signals are released. Consequently, this proposed ETS cuts down the dynamic process and reduces the transmission burden.
Next, seth=0.35. Under the improved static ETS with time-varying parameter σkt, the release instants and release interval are given in Figure 3. If we replace σkt with the constant σ, the corresponding release instants and release intervals are described in Figure 4. It is distinct that the transmission frequency at the beginning times in Figure 3 is higher than that in Figure 4.Table 1
The controller gainsK1 and K2 and trigger matrices Λ1 and Λ2.
t
0
1
…
20
Λ
1
1.88
−
0.02
−
0.12
1.90
1.33
0.95
0.40
7.23
…
1.77
0.13
0.13
6.14
Λ
2
2.50
0.07
0.07
2.40
2.13
0.95
0.40
1.85
…
1.97
−
0.22
−
0.03
5.14
K
1
−
0.9129
0.4357
1.1887
−
2.0034
…
−
1.4731
0.7982
K
2
−
0.9029
−
0.3271
0.1735
−
1.1657
…
1.1003
2.1372Figure 1
State trajectoryxt and semi-Markovian chain rt of the networked control system containing actuator faults and time-varying delay.
(a)
(b)Figure 2
The parameterσkt and the release instants.
(a)
(b)Figure 3
The release instants and release intervals of the improved static ETS.Figure 4
The release instants and release intervals of the ETS with constant parameterσ.Example 2.
Consider the following parameters of system (16):(88)A1=−200−0.9,A2=2.21.8−1.2−0.5,B1=1001,B2=1.11.5−11.4,K1=−100−1,K2=−100−1,λ¯i,j=0.8−0.8−0.50.5,β1=β2=0.2,σ=0.1.
Takeσ¯=0.3, σk1=0.01, ε=0.01, h=0.18, β1=0.2, and β2=0.2. Under different N, we obtain corresponding upper bounds of network time delay with τt,τ˙t∈ℋ. If N=0, then τM=1.2161. If N=1, then τM=1.7572. If N=2, then τM=2.2546. Table 2 shows the comparison of the upper bound of network time delay with other papers.
From Table2, we can see that when N=0, τM=1.2161 is bigger than other papers. The upper bound of τM increases as N increases. Thus, the conservativeness of the stability criterion decreases as N increases.Table 2
The value ofτM corresponding to different τ˙td2=−d1=d˙t.
Method\d˙t
0.1
0.2
0.3
0.8
N = 2
2.2546
2.1362
2.1041
1.6771
N = 1
1.7572
1.6035
1.4722
1.1095
N = 0
1.2161
0.9773
0.8554
0.6161
[45]
0.4945
0.4703
0.3634
0.3544
[10]
0.744
0.66
0.51
0.43


## 6. Conclusions

To sum up, event-triggered control problem for the networked control system with network delay and stochastic jumping parameters is investigated. The jumping among the parameters is subject to semi-Markovian jump process. More in combination with the actual situation, the actuator faults which also have the semi-Markovian jump property are considered. The feature of this paper is that an improved static ETS is proposed to change the trigger frequency at different stages. Consequently, the burden of transmission is reduced and the system dynamic process is shortened. A stability criterion with lower conservativeness is obtained with the help of applying the Bessel–Legendre inequalities approach and constructing an appropriate LKF. The criterion is indexed byN. The conservatism will decrease when N increases. Moreover, for this comprehensive system model, this paper designs an effective event-triggered controller. Finally, for verifying the validity of the results, numerical examples are presented.The further topics of the research can be that the investigation of the improved event-triggered filtering problem of the semi-Markovian jump networked control system containing actuator faults. The further improvement of ETS for some requirements of this comprehensive system.



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*Source: 6927528-2019-10-31.xml*