# H ∞ Guaranteed Cost Fault-Tolerant Control of Double-Fault Networked Control Systems: Piecewise Delay Method

Authors: Qixin Zhu; Kaihong Lu; Yonghong Zhu
Journal: Mathematical Problems in Engineering
 (2019)

---

## Abstract

The term double-fault networked control system means that sensor faults and actuator faults may occur simultaneously in networked control systems. The issues of modelling and anH∞ guaranteed cost fault-tolerant control in a piecewise delay method for double-fault networked control systems are investigated. The time-varying properties of sensor faults and actuator faults are modelled as two time-varying and bounded parameters. Based on the linear matrix inequality (LMI) approach, an H∞ guaranteed cost fault-tolerant controller in a piecewise delay method is proposed to guarantee the reliability and stability for the double-fault networked control systems. Simulations are included to demonstrate the theoretical results of the proposed method.


---

## Body

## 1. Introduction

Networked control systems (NCS) are frequently encountered in many fields of applications due to their suitable and flexible structure [1–9]. However, in practical NCS, there inevitably exists time delay and data packet dropouts because of the introduction of the communication network [10–13]. Sensor faults and actuator faults also occur easily at the device level because of its large scale and complicated structure [14], which can have a negative impact on the system, such as performance decline and instability. Thus, guaranteed cost and fault-tolerant control of networked control systems has become a new popular issue in the network control field.To achieve stability requirements concerning sensor faults or actuator faults, fault-tolerant control has been investigated in many works [15–26]. Based on the Lyapunov stability theorem, a methodology for the design of fault-tolerant control systems for chemical plants with distributed interconnected processing units was presented by N. H. El-Farra and A. P. D. Gani [15]. Z. Qu and C. M. Ihlefeld devised a fault-tolerant robust controller for a class of nonlinear uncertain systems considering possible sensor faults and developed a robust measure to identify the stability- and performance-vulnerable failures [16]. Based on the integrity control theory, a robust fault-tolerant controller for NCS with actuator faults was discussed by Y. N. Guo [17]. The diagnosis of actuator component faults and fault-tolerant control for a class of networked control systems using adaptive observer techniques was investigated in [18]. A switched model based on probability for NCS was proposed in [19] to research issues of fault-tolerant control when actuators age or become partially disabled.Recently, guaranteed cost control that can guarantee the stability of a system and make it meet a certain performance indicator has become popular in NCS [4–7, 27–29]. Stability guaranteed active fault-tolerant control against actuators failures in NCS was addressed by S. Li [27]. X. Y. Luo and M. J. Shang proposed the so-called guaranteed cost active fault-tolerant controller (AFTC) strategy in [28]. The issue of guaranteed cost reliable control with regional pole constraint against actuator failures was investigated by H. M. Soliman in [6]. In [7], X. Li investigated the issue of integrity against actuator faults for NCS under variable-period sampling, in which the existence conditions of a guaranteed cost fault-tolerant control law was tested in terms of the Lyapunov stability theory. The resilient reliable dissipativity performance index for systems including actuator faults and probabilistic time delay signals is investigated by authors in [30, 31]. Unfortunately, all the previously mentioned literature investigated the guaranteed cost fault-tolerant control for NCS of single-faults (just considering the condition that only actuator faults occur or sensor faults occur). Few articles examine the double-fault issue. In practical application, it is easy for the sensors and actuators to become faulty simultaneously when the NCS works in a poor environment and is affected by external disturbance. Improving the control performance of NCS when double-faults occur is important. This motivates us to investigate H∞ guaranteed cost fault-tolerant control of double-fault NCS, which is a necessary but challenge task.This paper develops aH∞ guaranteed cost fault-tolerant control for a double-fault NCS to guarantee its stability. The time-varying properties of sensor faults and actuator faults are modelled as two time-varying and bounded parameter matrices, and the networked control system is built as a linear closed-loop system with transmission delays and data packet dropouts. Different from [30, 31], here it is necessary to deal with two dynamic matrices. One of them is located at the left hand side of the gain matrix, and the other one is located at the right hand side of the gain matrix. This brings challenges to searching a feasible control gain. To solve such problem, a piecewise delay method is proposed to analyse the delay-dependent faulty system for reducing the conservatism. The delay falling in each subinterval is treated as a case. For different cases, different weighted technology is used to derive the H∞ guaranteed cost fault-tolerant condition, and the controller parameter for this NCS is obtained by solving several sets of LMIs. Compared with [26, 32–34], the delay considered in this paper can be continuously changed with time.Notation. Rn denotes the n-dimensional Euclidean space. The superscript “T” stands for matrix transposition. The notation X>0 means that the matrix X is a real positive definite matrix. I is the identity matrix of appropriate dimensions. XZ∗Y denotes a symmetric matrix, where*denotes the entries implied by symmetry.


## 2. Modelling of Double-Fault NCS

The linear control plant of NCS with uncertain parameters and external disturbance can be expressed as(1)x′t=A+ΔAxt+B+ΔBut+Hωtyt=Cxtwhere x(t)∈Rn, u(t)∈Rm, y(t)∈Rr, and ω(t)∈L2[0,∞)∈Rn represent state value, input, output, and external disturbance, respectively. Separately, A, B, H, and C are matrices with appropriate dimensions; ΔA and ΔB are matrices with uncertain time-varying parameters, satisfying [ΔA,ΔB]=DF(t)[E1,E2]; F(t) is an unknown matrix function with Legesgue measurable properties, satisfying FT(t)F(t)≤I; and D, E1, and E2 are constant matrices with appropriate dimensions.For the convenience of the later formulation, two concepts can be initially introduced.Definition 1.
The sensor faults and actuator faults could occur simultaneously, that is, there may be two types of faults at one time; this kind of fault is called adouble-fault.Definition 2.
The sensor faults and actuator faults do not occur simultaneously, that is, there is only one type of fault at one time; this kind of fault is called asingle-fault.The structure of a double-fault NCS is shown in Figure1. Transmission delays induced by the network are the sensor-to-controller delay τsc and the controller-to-actuator delay τca. These two delays can be combined when the feedback controller is static. The state of the system is assumed to be completely measurable. A piecewise continuous feedback controller, which is realized by a zero-order hold (ZOH), is employed: (2)ut=Kx~t-τca,t∈tk,tk+1,k=1,2,⋯where K is the state feedback gain matrix to be designed and tk is the sampling instant.Figure 1
The structure of double-fault NCS.Considering that sensor faults may occur,xiF(t) is used to represent the data from the i-th sensor. In this paper, we consider faults that include outage and loss of effectiveness. If the i-th sensor is an outage, the corresponding sampling work is interrupted, and the sampling data keeps the default value xiF(t)=0. If the i-th sensor loses effectiveness, the sampling data is inaccurate and nonzero. We denote the sensor fault model as(3)xiFt=gitxit,i=1,2,⋯,nwhere gi(t) is the time-varying sensor efficiency factor, gi=1 represents that the i-th sensor is normal, gi=0 represents that its fault is outage, and 0<gi<1 or gi>1 represents that its fault is loss of effectiveness. The upper bound of sensor efficiency factor gi(t) is denoted by a constant gui satisfying gui>1, while its lower bound is denoted by a constant gli satisfying 0≤gli<1.DenotingxFt=x1Ftx2Ft⋯xnFtT, we have(4)xFt=Gtxtwhere G(t)=diag(g1(t),g2(t),⋯,gn(t)) is the sensor fault indicator matrix. Correspondingly, its upper bound matrix is Gu=diag(gu1,gu2,⋯,gun), and its lower bound matrix is Gl=diag(gl1,gl2,⋯,gln).Data packet dropouts in NCS are also unavoidable because of limited bandwidth. Considering that data packet dropouts may occur, the network is modelled as a switch. When the switch is located inS1 position, the data packet containing x(tk) is transmitted, and the controller utilizes the updated data. When it is located in the S2 position, the data packet dropouts occur, and the controller uses the old data. For a fixed sampling period h, the dynamics of the switch can be expressed as follows:The NCS with no packet dropout at timetk: (5)x~t=Gtk-τscxtk-τscThe NCS with one packet dropout at time tk:(6)x~t=Gtk-τsc-hxtk-τsc-hThe NCS with dk∈Z+ packet dropout at time tk:(7)x~t=Gtk-τsc-dkhxtk-τsc-dkh.Because the feedback controller is static, (2) can be expressed as(8)ut=KGtk-τca-τsc-dkhxtk-τca-τsc-dkhConsidering that actuator faults may also occur,ujF(t) is used to represent the signal from the j-th actuator. Similarly, we denote the actuator fault model as(9)uiFt=δjtujt,j=1,2,⋯,mwhere δj(t) is the time-varying actuator efficiency factor, δj=1 represents that the j-th actuator is normal, δj=0 represents that its fault is an outage, and 0<δj<1 or δj>1 denotes that its fault is a loss of effectiveness. The upper bound of sensor efficiency factor δj(t) is denoted by a constant δuj satisfying δuj>1, while its lower bound is denoted by a constant δlj satisfying 0≤δlj<1.Denoting the faulty control signaluF(t)=u1F(t)u2F(t)⋯umF(t)T, we can obtain the fault-tolerant control law as(10)uFt=Θtut=ΘtKGtk-τca-τsc-dkhxtk-τca-τsc-dkhwhere Θ(t)=diag(δ1(t),δ2(t),⋯,δm(t)) is the actuator fault indicator matrix. Correspondingly, its upper bound matrix is Θu=diag(δu1,δu2,⋯,δum), and its lower bound matrix is Θl=diag(δl1,δl2,⋯,δlm).Letη(t)=t-tk+τca+τsc+dkh; t∈[tk,tk+1); (10) can now be expressed as follows:(11)uFt=Θtut=ΘtKGt-ηtxtk-ηtObviously, the delay part η(t) may vary with time t, and it satisfies(12)ηm≤ηt=t-tk+τca+τsc+dkh≤ηMt∈tk,tk+1From the upper boundsGu,Θu of fault indicator matrices and lower bounds Gl,Θl of fault indicator matrices, the nonsingular mean-value matrices are separately obtained as (13)G0=diagg01,g02,⋯,g0n,g0i=gui+gli2Θ0=diagδ01,δ02,⋯,δ0m,δ0j=δuj+δlj2Moreover, the following two time-varying matrices are introduced:(14)Lt-ηt=diagl1t-ηt,l2t-ηt,⋯,lnt-ηt,liηt=giηt-g0ig0iΓt=diagλ1t,λ2t,⋯,λmt,λit=δjt-δ0jδ0jObviously, we have(15)-1≤gli-g0ig0i≤lit-ηt=gi-g0ig0i≤gui-g0ig0i=gui-gligui+gli≤1Similarly, we have (16)-1≤λjt≤1In expressions from (13) to (16), i and j meet i=1,2,...,n, j=1,2,...,m. Based on (15) and (16), we have(17)-In×n≤Lt-ηt≤In×n-Im×m≤Γt≤Im×mFrom on (14), the following can be obtained: (18)git-ηt=g0i1+lit-ηt,i=1,2⋯,n;δit=δ0i1+λit,j=1,2⋯,m.Naturally, the time-varying fault indicator matrices can be rewritten as(19)Gt-ηt=G0I+Lt-ηtΘt=Θ0I+ΓtInserting (19) into (11), we have(20)ut=Θ0I+ΓtKG0I+Lt-ηtxt-ηtThen, the new model of double-fault NCS can be obtained as follows:(21)x′t=A+ΔAxt+B+ΔBΘ0I+ΓtKG0I+Lt-ηtxt-ηt+Hωtyt=CxtRemark 3.
The networked control systems in faulty case can be modelled as system (21) with the effects of time-varying delay η(t) whose upper bound and lower bound are described in (12). Unlike the previous models, [17, 21, 27, 35], this model is related to both sensor faults and actuator faults, the faults are time-varying, which are reflected by the time-varying parameters L(t-η(t)) and Γ(t). From (17), we undoubtedly know the time-varying parameter L(t-η(t)) satisfies LTL=L2≤I, while the parameter Γ(t) satisfies ΓTΓ=Γ2≤I.Remark 4.
From the descriptions of faults matrices, we undoubtedly know ifli=1/g0i-1 (i=1,2⋯,n), we can obtain gi=1 and G0(I+L(t-η(t)))=I, and then model (21) is an actuator fault model. Similarly, if λi=δj-δ0j/δ0j (j=1,2⋯,m), we can obtain δj=1 and Θ0(I+Γ(t))=I, and then model (21) represents a sensor fault model. Therefore, model (21) of a double-fault NCS contains cases of single-faults, and the single-faults are a special form of double-faults.
In the following section, a fundamental preliminary result is presented to guarantee the performance of a double-fault NCS based on the delay information.


## 3. Performance Analysis of Double-Fault NCS

For the system model (21) established in Section 2, the cost function is given as follows:(22)J=∫0∞xTtS1xt+uTtS2utdt=∫0∞xTtS1xt+Θ0I+ΓKG0I+Lxt-ηtTS2Θ0I+ΓKG0I+Lxt-ηtdtwhere S1 and S2 are symmetric positive definite matrices.Definition 5.
For model (21) and its cost function (22), if there exists a control gain matrix K satisfying the conditions
( 1 ) the closed-loop system is asymptotically stable when ω(t)=0;
( 2 ) for any zero initial condition and any nonzero vector ω(t)∈L2[0,∞), given γ>0, the output y(t) satisfies y(t)2≤γω(t)2;
( 3 ) there exists a constant J0, and the cost function defined as (22) satisfies J∞≤J0,
then matrixK is the H∞ guaranteed cost control gain of double-faults NCS.
To analyse the stability of the system expediently, the following lemmas are introduced.Lemma 6 (see [36, 37]).
For any matricesW, M, N, F(t) with FTF≤I, and any scalar ε>0, the inequality holds as (23)W+MFtN+NTFTtMT≤W+εMMT+ε-1NTNLemma 7 (see [38]).
Ifη1≤η(t)≤η2, for any matrices Π1, Π2, and Φ, the following inequalities are equivalent:
( 1 )  [η(t)-η1]Π1+[η2-η(t)]Π2+Φ<0;
( 2 )  [η2-η1]Π1+Φ<0,[η2-η1]Π2+Φ<0.The fundamental preliminary result is presented in the following theorem.Theorem 8.
Given symmetric positive definite matricesS1 and S2, a set of constant ηm, ηM, ρ1>0, ρ2>0, and α=ηM-ηm/2. If there exists a set of symmetric positive definite matrices Ri(i=1,2,3) and matrix P>0, as well as matrices Mlβ, M2β, Nlβ, N2β, Uβ (β=1,2,3,4,5,6,7,8), K, and a set of constants, ε>0 and γ>0, satisfying the LMIs(24)-εI000εD~T∗-S1+CTC-100I~∗∗-S2-10Σ~∗∗∗-ΓlΩlk∗∗∗∗Φ~l∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗→00000000000Θ0KG0Θ0000000E~TΘ~I~0ΔK~T-εIE2Θ0KG00E2Θ00∗-ρ1U500KG0T∗∗-ρ1-1U5-100∗∗∗-ρ2I0∗∗∗∗-ρ2-1I<0l=1,2;k=1,2where(25)Φ~1=-R1+U1A+U1ATR1+U2AT+M11U3AT-N11U4AT∗-R1+M12+M12TM13T-N12M14T∗∗-R3α-N12-N12TR3α-N14T∗∗∗-R3α∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗→P-U1+U5ATΨ~1+U6ATU7AT+U8AT+U1H0-U2+M15TΨ~2+M16TU2H+M17T+M18T0-U3-N15TΨ~3-N16TU3H-N17T-N18T0-U4Ψ~4U4H0ηm2R1+αR2+R3-U5-U5TΨ~5-U6TU5H-U7T-U8T0∗Ψ~6Ξ0∗∗U7H+U8H+U7HT+U8HT0∗∗∗-γ2IΦ~2=-R1+U1A+U1ATR1+U2ATU3AT+M21U4AT-N21∗-R1-R2αR2α+M22-N22∗∗-R2α+M23+M23TM24T-N23∗∗∗-N24-N24T∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗→P-U1+U5ATΨ~1′+U6ATU7AT+U8AT+U1H0-U2Ψ~2′U2H0-U3+M25TΨ~3′+M26TU3H+M27T+M28T0-U4-N25TΨ~4′-N26TU4H-N27T-N28T0ηm2R1+αR2+R3-U5-U5TΨ~5′-U6TU5H-U7T-U8T0∗Ψ~6′Ξ′0∗∗U7H+U8H+U7HT+U8HT0∗∗∗-γ2IΓ1=αR2,Γ2=αR3,Ω11=αM1T,Ω12=αN1T,Ω21=αM2T,Ω22=αN2T,M1T=M11TM12TM13TM14TM15TM16TM17TM18T;M2T=M21TM22TM23TM24TM25TM26TM27TM28T;N1T=N11TN12TN13TN14TN15TN16TN17TN18T;N2T=N21TN22TN23TN24TN25TN26TN27TN28T;Ξ=U7BΘ0KG0T+U8BΘ0KG0T+U6H-M17T-M18T+N17T+N18T;Ψ~i=UiBΘ0KG0+N1i-M1i,i=1,2,3,4,5;Ψ~6=U6BΘ0KG0+U6BΘ0KG0T-M16-M16T+N16+N16T;Ξ′=U7BΘ0KG0T+U8BΘ0KG0T+U6H-M27T-M28T+N27T+N28T;Ψ~i′=UiBΘ0KG0+N2i-M2i,i=1,2,3,4,5;Ψ~6′=U6BΘ0KG0+U6BΘ0KG0T-M26-M26T+N26+N26T;D~=U1DTU2DTU3DTU4DTU5DTU6DTU7DT+U8DT0T;E~=E10000E2Θ0KG000;I~0=00000I00T;I~=I0000000;Θ~=U1BΘ0KG0TU2BΘ0KG0TU3BΘ0KG0TU4BΘ0KG0TU5BΘ0KG0TU6BΘ0KG0TU7BKG0T+U8BKG0T0T;Δ=U1BΘ0TU2BΘ0TU3BΘ0TU4BΘ0TU5BΘ0TU6BΘ0TU7BΘ0T+U8BΘ0T0T;K~=00000KG000;Σ~=00000Θ0KG000,then model (21) is asymptotically stable with the H∞ norm bound γ. In addition, the upper bound J0 of cost function J is given as (26)J0=xT0tPx0+ηm∫-ηm0∫s0x′TτR1x′τdsdτ+∫-η1-ηm∫s0x′TτR2x′τdsdτ+∫-ηM-η1∫s0x′TτR3x′τdsdτProof.
First, with the definition ofα=ηM-ηm/2 and η1=ηm+α, the interval of delay is distributed into two subintervals as follows:(27)ηt∈ηm,ηM=ηm,η1∪η1,ηM.Then, we consider the Lyapunov-Krasovskii functional as follows:(28)vt=xTtPxt+ηm∫t-ηmt∫stx′TτR1x′τdsdτ+∫t-η1t-ηm∫stx′TτR2x′τdsdτ+∫t-ηMt-η1∫stx′TτR3x′τdsdτwhere matrix P satisfies P>0 and Ri (i=1,2,3) are symmetric positive definite matrices with appropriate dimensions. For the convenience of writing, we denote L=L(t-η(t)) and Γ=Γ(t) in the following expressions. Calculating the derivative of Lyapunov-Krasovskii function and based on (21), we have(29)v′t=2xTtPx′t+η2mx′TtR1x′t-ηm∫t-ηmtx′TsR1x′sds+η1-ηmx′TtR2x′t-∫t-η1t-ηmx′TsR2x′sds+ηM-η1x′TtR3x′t-∫t-ηMt-η1x′TsR3x′sds+2εTtωTtUA+ΔAxt+B+ΔBΘ0I+ΓKG0I+Lxt-ηt+Hωt-x′twhere v′(t)=limsupδ→0+1/δvt+δ-vt [39].
Based on Jessen’s inequality, we have(30)-ηm∫t-ηmtx′TsR1x′sds≤xTt,xTt-ηm-R1R1R1-R1xtxt-ηm(31)-∫t-η1t-ηmx′TsR2x′sds≤1αxTt-ηm,xTt-η1-R2R2R2-R2xt-ηmxt-η1(32)-∫t-ηMt-η1x′TsR3x′sds≤1αxTt-η1,xTt-ηM-R3R3R3-R3xt-η1xt-ηMwhere α=ηM-ηm/2. For the convenience of the following discussion, we define (33)εTt=xTt,xTt-ηm,xTt-η1,xTt-ηMx′Tt,xTt-ηt,ωTt;Case  1. If η(t)∈[ηm,η1], weighted technology based on the principle of Newton-Leibniz is introduced as follows:(34)2εTtωTtM1xt-ηm-xt-ηt-∫t-ηtt-ηmx′Tsds=0and(35)2εTtωTtN1xt-ηt-xt-η1-∫t-η1t-ηtx′Tsds=0Because R2>0, we have(36)-2εTtωTtM1∫t-ηtt-ηmx′sds≤ηt-ηmεTtωTtM1R2-1M1TεTtωTtT+∫t-ηtt-ηmx′TsR2x′sdsand(37)-2εTtωTtN1∫t-η1t-ηtx′sds≤η1-ηtεTtωTtN1R2-1N1TεTtωTtT+∫t-η1t-ηtx′TsR2x′sdsTherefore,(38)-2εTtωTtM1∫t-ηtt-ηmx′sds-2εTtωTtN1∫t-η1t-ηtx′sds≤ηt-ηmεTtωTtM1R2-1M1TεTtωTtT+η1-ηtεTtωTtN1R2-1N1TεTtωTtT+∫t-η1t-ηmx′TsR2x′sdsIn addition, yT(t)y(t)-γ2ωTtωt both on the left and right sides of equality (29) and xT(t)S1x(t)+[Θ0(I+Γ)KG0(I+L)x(t-η(t))]TS2Θ0(I+Γ)KG0(I+L)x(t-η(t)) on the right of the equal sign “=”, and then inserting (30), (32), (34), (35), (38) to the obtained inequality, we have(39)v′t+yTtyt-γ2ωTtωt≤εTtωTtΦ1εTtωTtT+ηt-ηmεTtωTtM1R2-1M1TεTtωTtT+η1-ηtεTtωTtN1R2-1N1TεTtωTtTwhere(40)Φ1=Ξ1R1+U2A+ΔAT+M11U3A+ΔAT-N11U4A+ΔAT∗-R1+M12+M12TM13T-N12M14T∗∗-R3α-N12-N12TR3α-N14T∗∗∗-R3α∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗→P-U1+U5A+ΔATΨ1+U6A+ΔATU7A+ΔAT+U8A+ΔAT+U1H0-U2+M15TΨ2+M16TU2H+M17T+M18T0-U3-N15TΨ3-N16TU3H-N17T-N18T0-U4Ψ4U4H0ηm2R1+αR2+R3-U5-U5TΨ5-U6TU5H-U7T-U8T0∗Ψ6Ξ20∗∗U7H+U8H+U7HT+U8HT0∗∗∗-γ2IΞ1=-R1+U1A+ΔA+U1A+ΔAT+CTC+S1,Ξ2=U7B+ΔBΘ0I+ΓKG0I+LT+U8B+ΔBΘ0I+ΓKG0I+LT+U6H-M17T-M18T+N17T+N18T,Ψi=UiB+ΔBΘ0I+ΓKG0I+L+N1i-M1i,i=1,2,3,4,5;Ψ6=U6B+ΔBΘ0I+ΓKG0I+L+U6B+ΔBΘ0I+ΓKG0I+LTΘ0I+ΓKG0I+LTS2Θ0I+ΓKG0I+L-M16-M16T+N16+N16Tif(41)Φ1+ηt-ηmM1R2-1M1T+η1-ηtN1R2-1N1T<0
Next, we need to acquire the inequality (24) through a transformation based on inequality (41), which is equivalent to the following inequalities by applying the theory given in Lemma 7 and the Schur complement, also used by D. Yue [38] in the previous study:(42)-η1-ηm-1R2M1T∗Φ1<0(43)-η1-ηm-1R2N1T∗Φ1<0Premultiplying and postmultiplying the inequalities above by diag((η1-ηm)I,I), we have (44)-αR2Ω1k∗Φ1<0Applying the theory of the Schur complement to inequality (44), we have(45)-S1+CTC-100I~∗-S2-10Σ∗∗-αR2Ω1k∗∗∗Φ1′<0where(46)Φ1′=Ξ1′R1+U2A+ΔAT+M11U3A+ΔAT-N11U4A+ΔAT∗-R1+M12+M12TM13T-N12M14T∗∗-R3α-N12-N12TR3α-N14T∗∗∗-R3α∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗→P-U1+U5A+ΔATΨ1+U6A+ΔATU7A+ΔAT+U8A+ΔAT+U1H0-U2+M15TΨ2+M16TU2H+M17T+M18T0-U3-N15TΨ3-N16TU3H-N17T-N18T0-U4Ψ4U4H0ηm2R1+αR2+R3-U5-U5TΨ5-U6TU5H-U7T-U8T0∗Ψ6′Ξ20∗∗U7H+U8H+U7HT+U8HT0∗∗∗-γ2IΞ1′=-R1+U1A+ΔA+U1A+ΔATΨ6′=U6B+ΔBΘ0I+ΓKG0I+L+U6B+ΔBΘ0I+ΓKG0I+LT-M16-M16T+N16+N16T,Σ=00000Θ0I+ΓKG0I+L00.Inequality (44) can be written as (47)
-
S
1
+
C
T
C
-
1
0
0
I
~
∗
-
S
2
-
1
0
Σ
∗
∗
-
α
R
2
Ω
1
k
∗
∗
∗
Φ
1
′
′
+
0
0
0
D
~
F
0
0
0
E
^
T
T
+
0
0
0
E
^
T
F
T
0
0
0
D
~
T
<
0where(48)Φ1″=-R1+U1A+U1ATR1+U2AT+M11U3AT-N11U4AT∗-R1+M12+M12TM13T-N12M14T∗∗-R3α-N12-N12TR3α-N14T∗∗∗-R3α∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗→P-U1+U5ATΨ1″+U6ATU7AT+U8AT+U1H0-U2+M15TΨ2″+M16TU2H+M17T+M18T0-U3-N15TΨ3″-N16TU3H-N17T-N18T0-U4Ψ4′U4H0ηm2R1+αR2+R3-U5-U5TΨ5″-U6TU5H-U7T-U8T0∗Ψ6″Ξ2″0∗∗U7H+U8H+U7HT+U8HT0∗∗∗-γ2IΞ2″=U7BΘ0I+ΓKG0I+LT+U8BΘ0I+ΓKG0I+LT+U6H-M17T-M18T+N17T+N18T,Ψi″=UiBΘ0I+ΓKG0I+L+N1i-M1i,i=1,2,3,4,5;Ψ6′=U6BΘ0I+ΓKG0I+L+U6BΘ0I+ΓKG0I+LT-M16-M16T+N16+N16TE^=E10000E2Θ0I+ΓKG0I+L00.With the definition of scalar ε>0, we apply the theory given in Lemma 6 to (47), also used by Y. Wang and L. Xie [37], in which the uncertain matrix F can be eliminated, and a sufficient condition of (47) is obtained. (49)-S1+CTC-100I~∗-S2-10Σ∗∗-αR2Ω1k∗∗∗Φ1′′+ε0D0D~0D0D~T+ε-1000E^T000E^TT<0Applying the Schur complement to inequality (49), we have(50)-εI000εD~T0∗-S1+CTC-100I~0∗∗-S2-10Σ0∗∗∗-αR2Ω1k0∗∗∗∗Φ1′′E^T∗∗∗∗∗-εI<0There exists ρ1>0, ρ2>0. Based on (17) and Remark 3 in Section 2, we know that LTL≤I and ΓTΓ≤I. According to expressions (47), (49), and (50), we obviously know that U5>0. Using Lemma 6 again, we have(51)Θ0I+ΓKG0L+Θ0I+ΓKG0LT≤ρ1-1Θ0I+ΓKG0U5-1Θ0I+ΓKG0T+ρ1U5and(52)Θ0ΓKG0+Θ0ΓKG0T≤ρ2-1Θ0Θ0T+ρ2KG0TKG0Based on inequalities (51), (52) and the Schur complement, we know inequality (24) is a sufficient condition of inequality (50), while inequality (50) is equivalent to inequality (41). Therefore, we can undoubtedly obtain inequality (24) as a sufficient condition of inequality (41). Thus, based on inequality (39) and (41), we know
( 1 ) if ω(t)≡0, obviously, we have v′(t)<0, so system (21) is asymptotically stable;
( 2 ) if x(0)≡0, we know v(0)=0. In addition, it can be obtained that v(∞)≥0.
Therefore,(53)∫0∞v′t+yTtyt-γ2ωTtωtdt=v∞+∫0∞yTtyt-γ2ωTtωtdt<0Thus, (54)∫0∞yTtyt<∫0∞γ2ωTtωtdtBecause ω(t)∈L2[0,∞), we have(55)yt≤γωtT.It is known that model (21) is asymptotically stable with the H∞ norm bound γ.
Moreover, according to (39) and (41), we have(56)v′t≤-xTtS1xt-Θ0I+ΓKG0I+Lxt-ηtTS2Θ0I+ΓKG0I+Lxt-ηtThrough the integral operation, it can be determined that J≤v(0). In addition, by inserting t=0 into the Lyapunov-Krasovskii function shown as expression (28), the upper bound of cost function J can be obtained and shown as expression (26). Therefore, the theorem is verified if l=1.
Case 2. If η(t)∈[η1,ηM], weighted technology based on the principle of Newton-Leibniz is introduced as follows:(57)2εTtωTtM2xt-η1-xt-ηt-∫t-ηtt-η1x′Tsds=0(58)2εTtωTtN2xt-ηt-xt-ηM-∫t-ηMt-ηtx′Tsds=0Because R3>0, we have(59)-2εTtωTtM2∫t-ηtt-η1x′sds≤ηt-η1εTtωTtM2R3-1M2TεTtωTtT+∫t-ηtt-η1x′TsR3x′sds(60)-2εTtωTtN2∫t-ηMt-ηtx′sds≤ηM-ηtεTtωTtN2R3-1N1TεTtωTtT+∫t-ηMt-ηtx′TsR3x′sdsTherefore,(61)-2εTtωTtM2∫t-ηtt-η1x′sds-2εTtωTtN2∫t-ηMt-ηtx′sds≤ηt-η1εTtωTtM2R3-1M2TεTtωTtT+ηM-ηtεTtωTtN2R3-1N2TεTtωTtT+∫t-ηMt-η1x′TsR3x′sdsIn addition, yT(t)y(t)-γ2ωT(t)ω(t) both on the left and right sides of equality (29) and xT(t)S1x(t)+[Θ0(I+Γ)KG0(I+L)x(t-η(t))]TS2Θ0(I+Γ)KG0(I+L)x(t-η(t)) on the right side of equal sign “=”. By inserting (30), (31), (57), (58), and (61) into the obtained inequality, the item -∫t-ηMt-η1x′TsR3x′(s)ds can be offset, while the item -∫t-η1t-ηmx′TsR2x′(s)ds is offset in Case 1. Then, in the same methods of transformation as Case 1, the inequality (24) when l=2 can be obtained. Therefore, the proof is complete.Remark 9.
In the two different cases, a variational weighting matrix and Jessen’s inequalities are used to derive theH∞ guaranteed cost fault-tolerant condition of the system, in which more delay information is employed to reduce the conservatism.The next section will provide sufficient conditions for designing the guaranteed cost fault-tolerant control for a double-fault NCS.


## 4. Guaranteed Cost Fault-Tolerant Control of Double-Fault NCS

Inequality (24) is not linear with respect to the gain matrices of the controller, so it is needs to be reformulated into LMIs via a change of variables.Theorem 10.
Given symmetric positive definite matricesS1 and S2, a set of constant ηm, ηM, ρ1>0, ρ2>0, λi (i = from 1 to 7), and α=ηM-ηm/2. If there exists a set of symmetric positive definite matrices R~j(j=1,2,3), X, and matrix P~>0, as well as matrices M~1β, M~2β, N~1β, N~2β, (β=from 1 to 8), Y, and a set of constants, ε>0 and μ>0, satisfying the LMIs(62)-εI000εD-T∗-S1+CTC-100Π∗∗-S2-10Σ-∗∗∗-Γ-lΩ-lk∗∗∗∗Φ-l∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗00000000000Θ0Y0Θ0000000E-TΘ-X-Δ-Y-T-εIE2Θ0Y0E2Θ00∗-ρ1XT00YT∗∗-ρ1-1X00∗∗∗-ρ2I0∗∗∗∗-ρ2-1I<0l=1,2;k=1,2where(63)Φ˘1=-R~1+λ1AXT+λ1XATR~1+λ2XAT+M~11λ3XAT-N~11λ4XAT∗-R~1+M~12+M~12TM~13T-N~12M~14T∗∗-R~3α-N~12-N~12TR~3α-N~14T∗∗∗-R~3α∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗→P~-λ1XT+XATΨ-1+λ5XATλ6XAT+λ7XAT+λ1HXT0-λ2XT+M~15TΨ-2+M~16Tλ2HXT+M~17T+M~18T0-λ3XT-N~15TΨ-3-N~16Tλ3HXT-N~17T-N~18T0-λ4XTΨ-4λ4HXT0ηm2R~1+αR~2+R~3-XT-XΨ-5-λ5XHXT-λ6X-λ7X0∗Ψ-6Ξ-0∗∗λ6HXT+λ7HXT+λ6XHT+λ7XHT0∗∗∗-μIΦ~2=-R~1+λ1AXT+λ1XATR~1+λ2XATλ3XAT+M~21λ4XAT-N~21∗-R~1-R~2αR~2α+M~22-N~22∗∗-R~2α+M~23+M~23TM~24T-N~23∗∗∗-N~24-N~24T∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗∗→P~-λ1XT+XATΨ-1′+λ5XATλ6XAT+λ7XAT+λ1HXT0-λ2XTΨ-2′λ2HXT0-λ3XT+M~25TΨ-3′+M~26Tλ3HXT+M~27T+M~28T0-λ4XT-N~25TΨ-4′-N~26Tλ4HXT-N~27T-N~28T0ηm2R~1+αR~2+R~3-X-XTΨ-5′-λ5XHXT-λ6X-λ7X0∗Ψ˘6′Ξ′-0∗∗λ6HXT+λ7HXT+λ6XHT+λ7XHT0∗∗∗-μIΓ1=αR2Γ-1=αR~2,Γ2=αR˘3,Ω-11=αM~1T,Ω-12=αN~1T,Ω-21=αM~2T,Ω˘22=αN~2T,M~1T=M~11TM~12TM~13TM~14TM~15TM~16TM~17TM~18T;M~2T=M~21TM~22TM~23TM~24TM~25TM~26TM~27TM~28T;N~1T=N~11TN~12TN~13TN~14TN~15TN~16TN~17TN~18T;N2T=N~21TN~22TN~23TN~24TN~25TN~26TN~27TN~28T;Ξ-=λ6BΘ0YT+λ7BΘ0YT+λ5HXT-M~17T-M~18T+N~17T+N~18T;Ψ-i=λiBΘ0Y+N~1i-M~1i,i=1,2,3,4;Ψ-5=BΘ0Y+N~1i-M~1i,Ψ-6=λ5BΘ0Y+λ5BΘ0YT-M-16-M-16T+N-16+N-16T;Ξ′-=λ6BΘ0YT+λ7BΘ0YT+λ5HXT-M~27T-M~28T+N~27T+N~28T;Ψ-i′=λiBΘ0Y+N~2i-M~2i,i=1,2,3,4;Ψ-5′=BΘ0Y+N~2i-M~2i;Ψ-6′=λ5BΘ0Y+λ5BΘ0YT-M~26-M~26T+N~26+N~26T;D-=λ1DTλ2DTλ3DTλ4DTDTλ5DTλ6DT+λ7DT0T;E-=E1XT0000E2Θ0Y00;X-=00000XT00T;Π=XT0000000;Θ-=λ1BΘ0YTλ2BΘ0YTλ3BΘ0YTλ4BΘ0YTBΘ0YTλ5BΘ0YTλ6BYT+λ7BYT0T;Δ-=λ1BΘ0Tλ2BΘ0Tλ3BΘ0Tλ4BΘ0TBΘ0Tλ5BΘ0Tλ6BΘ0T+λ7BΘ0T0T;Y-=00000Y00;Σ-=00000Θ0Y00,then the H∞ guaranteed cost control gain K=YX-TG0-1 can render model (21) to be asymptotically stable with the H∞ norm bound γ=μ. The upper bound J0 of cost function J is given as(64)J0=xT0X-1P~X-Tx0+ηm∫-ηm0∫s0x′TτX-1R~1X-Tx′τdsdτ+∫-η1-ηm∫s0x′TτX-1R~2X-Tx′τdsdτ+∫-ηM-η1∫s0x′TτX-1R~3X-Tx′τdsdτ.Proof.
The proof is based on a suitable transformation and a change of variables allowing us to obtain inequality (24) in Theorem 8. First, we define U5=U0, Ui=λiU0 (i= from 1 to 4), U6=λ5U0 in (24). Because we consider the dimension of statexis equal to that of outside disturbance ω in this paper, we can also define U7=λ6U0, U8=λ7U0. Obviously, (24) implies U5>0, so U0 is nonsingular. Then, using the analysis method of D. Yue [40] and Z. Wang [41], pre- and postmultiplying both sides of inequality (24) with diag(I,I,I,X,X~,I,X,I,I,I) and its transpose, where X~=diag(X,X,X,X,X,X,X,X) and X=U0-1, introducing new variables XPXT=P~; XRjXT=R~j (j=1,2,3); XMlβXT=M~lβ, XNlβXT=N~lβ (l=1,2;β=1,2,⋯,8); KG0XT=Y and μ=γ2. From the definition of G0, we know G0 is invertible, so K can be obtained by calculating K=YX-TG0-1. It is easy to see that (62) and (64) respectively imply (24) and (26). Therefore, from Theorem 8, we can complete the proof.To obtain the optimal boundJ∗ shown in (26), the commonly used method is to consider it as an optimization problem like [42], in which the expression of initial state x(t) (t∈[-ηM,0]) needs to be known. The expression of initial state is not given in this paper. Therefore, the optimization method used in [42] cannot be used here. A practical method to obtain J∗, also used by D. Yue [43], is employed as follows.Supposex′(t) is bounded if t∈[-ηM,0] and satisfying x′Tτx′(τ)≤l. In addition, suppose that there exists βi>0(i=1,2,3), satisfying(65)X-1R~iX-T≤βiIi=1,2,3X-1P~X-T≤β4IInserting this into (64), we have(66)J0≤β4xT0x0+0.5lβ1ηm3+0.5lβ2η12-ηm2+0.5lβ3ηM2-η12=J∗Applying the Schur complement to the inequalities above, we have(67)-βiIX-1∗-Ri-1<0i=1,2,3-β4IX-1∗-P-1<0Then, combining (62) and (67), R~i-1,R~i (i=1,2,3), P~-1,P~,X-1,X exist simultaneously. We cannot directly use the LMI tools to solve the problem. Defining R~i-1=R^i (i=1,2,3), P~-1=P^, X-1=X^ and using the idea of the cone complementary linearization algorithm, the guaranteed cost fault-tolerant controller of system (21) and the value of optimal performance indicator J∗ can be obtained in the following method:(68)Minimize:traceR~1R^1+R~2R^2+R~3R^3+P~P^+X~X^+β4+0.5β1ηm3+0.5β2η12-ηm2+0.5β3ηM2-η12Subject  to:Inequalities62,-βiIX^∗-R^i<0i=1,2,3,-β4IX^∗-P^<0,XI∗X^≥0,R^1I∗R~1≥0,R^2I∗R~2≥0,R^3I∗R~3≥0,P^I∗P~≥0,R~i>0i=1,2,3.


## 5. Simulations

Example 11.
Consider inverted pendulum model that can usually be modelled as (1), and the system parameters are given as follows:(69)A=-21.1104,B=0.1-4,C=-0.31510.10.110.3,H=0.15-0.56-0.10.32,ΔA=0.04sin⁡t0.15sin⁡t-0.2sin⁡t0.75sin⁡t,ΔB=0.02sin⁡t-0.1sin⁡t,D=0.1-0.5,Ft=sin⁡t.Therefore, we have (70)E1=0.41.5,E2=0.2.For this simulation, the initial state of system is assumed x(0)=2-1T, and the external disturbance is considered as ω(t)=0.15-0.5T3s≤t≤4s0others. Here, we take the upper bound of time-varying delay as ηM=0.3s and its lower bound as ηm=0s, namely, 0≤η(t)≤0.3. In addition, the fault bounds of the system are given in Table 1.
We choose the parameters as follows:(71)λ1=0.1,λ2=0.15,λ3=-0.64,λ4=0.3,λ5=λ6=0.35,λ7=1.8,S2=0.31,ρ1=3000,ρ2=220.32,l=1,S1=100.130092.07.
By taking advantage of the LMI tool box and inserting the above parameters into inequalities (62) and (68), we can obtain the H∞ guaranteed cost control gain(72)K=YX-TG0-1=8.516315.1398with γ=μ=195.2516.
The corresponding optimal performance indicator (the upper bound value of guaranteed cost function) isJ∗=8316.2563.
The time-varying delay is shown in Figure2. In Figure 3, (a) is the actuator fault, which is a piecewise-linear function. It keeps the minimum value from 1.3s to 1.7s, while it keeps the maximum value from 3.7s to 4.3s. The sensor faults are shown as (b), which is sinusoidal. It should be noted that the green dotted line represents the fault of sensor 1, while the blue solid line represents the fault of sensor 2. Through the state response of the double-fault NCS shown in Figure 4 and corresponding control signal shown in Figure 5, we know the H∞ guaranteed cost controller designed in this paper is able to make the double-fault NCS asymptotically stable. The system gets preliminarily steady at 2s, and its state can return to the equilibrium position in a certain period of time when the NCS is affected by external disturbance. Compared with the state response of worse stability shown in Figure 6 when the method proposed in [28] is used for this double-fault problem, it sufficiently proves the effectiveness and feasibility of the method proposed in this paper.
To better illustrate the effectiveness of the method proposed in this paper, the following example is presented.Table 1
The bounds of faults.
Symbol Upper bound Lower bound ActuatorfaultsΘ 1.36 0.09 SensorFaultsG 1.65 0 0 1.75 0 0 0 0.15Figure 2
The time-varying delay in double-fault NCS.Figure 3
The time-varying faults.
(a)
Time-varying actuator fault (b)
Time-varying sensor faultFigure 4
The state response curve of double-fault NCS.Figure 5
The control input of double-fault NCS.Figure 6
The state response curve of double-fault NCS.Example 12.
Consider the parameters of system (1) as follows:(73)A=0.2100.3510-5.3-5.863.233.65-1.1-1.56-0.8900-1.58-2.85,B=5.591.2-0.891.3T,C=1.70.20.15-0.18,ΔA=0.82sin⁡t00000.12sin⁡t0000-0.58sin⁡t00001.01sin⁡t,ΔB=-0.15sin⁡t0.08sin⁡t0-0.28sin⁡t,H=-0.410.960.5200.06-0.5700.211.2100.590-0.32-0.11-0.35-0.03,D=I4×4,Ft=diagsin⁡t,sin⁡t,sin⁡t,sin⁡t.Therefore, we have (74)E1=diag0.82,0.12,-0.58,1.01,E2=-0.150.080-0.28T.For this simulation, the initial state of system is assumed as x(0)=1.21-0.510.23-0.18T, and the external disturbance is considered as(75)ωt=0.15cos⁡t0.15cos⁡t-0.12sin⁡t-0.25cos⁡tT6s≤t≤8s0others.
The uncertain time-varying delay satisfies0≤η(t)≤0.23s. In addition, the fault bounds of the system are given in Table 2. Other parameters are selected as follows:(76)λ1=1.2,λ2=-2.03,λ3=1.68,λ4=-2.3,λ5=-1.05,λ6=6.05,λ7=5.12,S1=diag198.88,156.34,87.09,128.76,S2=5.86,ρ1=57.98,ρ2=18.79,l=1.
By taking advantage of the LMI tool box and submitting these parameters above into inequalities (62) and (68), we can obtain the H∞ guaranteed cost control gain(77)K=YX-TG0-1=-1.72561.5512-0.85710.7451withγ=μ=581.9806.The corresponding optimal performance indicator is J∗=6502.1047. From the state response shown in Figure 7 and control signal shown in Figure 8, we undoubtedly know the double-fault NCS is asymptotically stable when the H∞ guaranteed cost controller is used. This further demonstrates the feasibility and effectiveness of the method proposed in this paper.Table 2
The bounds of faults.
Symbol Upper bound Lower bound ActuatorfaultsΘ 1.50 0.12 SensorFaultsG 1.85 0 0 0 0 1.60 0 0 0 0 1.25 0 0 0 0 1.65 0.25 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0.15Figure 7
The state response curve of double-fault NCS.Figure 8
The control input of double-fault NCS.


## 6. Conclusions

The issues of modelling andH∞ guaranteed cost fault-tolerant control of double-fault networked control systems have been addressed. The closed-loop model of a double-fault NCS is set up with regard to the influences of transmission delay, packet dropout, uncertain parameters, and external disturbance. In addition, the piecewise delay method is proposed to reduce the conservatism when analysing the delay-dependent faulty system. With the help of Lee Y S’s lemma, the sufficient condition of guaranteed cost fault-tolerant for time-varying double-fault NCS is introduced using the Lyapunov-Krasovskii theory and weighted technology. The method of designing a guaranteed cost fault-tolerant controller for this NCS is given based on LMI. Our next research task will be choosing more reasonable values of parameters λi (i=1,2,3,4,5,6,7) to reduce the conservatism further. Of course, the study on scheduling policy of double-fault NCS is also a challenge but indispensable work.



---

*Source: 6348727-2019-01-03.xml*