# Guaranteed Cost Fault-Tolerant Control for Networked Control Systems with Sensor Faults

Authors: Qixin Zhu; Kaihong Lu; Guangming Xie; Yonghong Zhu
Journal: Mathematical Problems in Engineering
 (2015)

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## Abstract

For the large scale and complicated structure of networked control systems, time-varying sensor faults could inevitably occur when the system works in a poor environment. Guaranteed cost fault-tolerant controller for the new networked control systems with time-varying sensor faults is designed in this paper. Based on time delay of the network transmission environment, the networked control systems with sensor faults are modeled as a discrete-time system with uncertain parameters. And the model of networked control systems is related to the boundary values of the sensor faults. Moreover, using Lyapunov stability theory and linear matrix inequalities (LMI) approach, the guaranteed cost fault-tolerant controller is verified to render such networked control systems asymptotically stable. Finally, simulations are included to demonstrate the theoretical results.


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## Body

## 1. Introduction

Feedback control systems wherein the control loops are closed through a real-time network are called networked control systems (NCS) [1]. Due to their suitable and flexible structure, NCS is frequently encountered in practice for such fields as information technology, life science, and aeronautical and space technologies. However, there exist not only induced delay, data packet loss, and sequence disordering in NCS, but also actuators or sensors faults, which could cause negative impact on the performance of the system and even lead to system instability. Recently, the fault-tolerant control of NCS has become a new popular issue in the control field [2–9].A fault-tolerant control algorithm for networked control systems is proposed based on Lyapunov stability theorem by Zheng and Fang [2]. Qu et al. have devised a fault-tolerant robust control for a class of nonlinear uncertain systems with possible sensor faults considered and developed a robust measure to identify the stability- and performance-vulnerable failures [3]. The faults of each sensor or actuator were taken as occurring randomly by Tian et al. [5], and their failure rates are governed by two sets of unrelated random variables satisfying certain probabilistic distribution. A sufficient condition is given by Zhang et al. [7], which could guarantee the stability of NCS with sensor failures or actuator failures and guarantee robustness to parameter uncertainties, but the issue of guaranteed cost is not discussed. A robust fault-tolerant control based on the integrity control theory when actuator faults occur is discussed by Zhang et al. [9]. Wang et al. [10] investigated the issue of integrity against actuator faults for NCS under variable-period sampling, in which the existence conditions of guaranteed cost faults-tolerant control law are testified in terms of Lyapunov stability theory, but not referring to the effects of uncertain parameters.Almost all literatures above consider the faults in some special cases. However, in practical application, because of large scale and complicated structure of NCS, the faults could vary from time to time when the system works in a poor environment. It is of great importance to explore a reasonable control method to guarantee the performance of NCS when time-varying faults occur. This motivates us to conduct the research work.Based on the network transmission environment and sampling theory, the networked control systems are firstly modeled as a discrete-time closed-loop system by considering the time-varying transmission delay and sensor faults simultaneously. The model of NCS is related to the boundary values of the sensor faults. Using Lyapunov stability theory, a sufficient condition is given, which can render the closed-loop NCS asymptotically stable and can guarantee it to meet the requirements of performance indicator (the upper bound of cost function). Based on LMI, the method of designing guaranteed cost fault-tolerant controller of NCS with time-varying faults is proposed in this paper.


## 2. Modeling of Networked Control Systems with Sensor Faults

A typical structure of NCS is shown in Figure1.Figure 1
The structure of networked control systems.In Figure1, τsc represents the transmission delay from sensor to the controller, while τca represents that from controller to the actuator. When the controller is static, the induced-delay time of system can be lumped as τ=τsc+τca.A linear control plant is described by state equation as follows:(1)x˙t=Aoxt+Boutyt=Coxt+Dout,where x∈Rn, u∈Rm, and y∈Rr represent state, input, and output vectors separately, while Ao,  Bo,  Co, and Do are matrices with appropriate dimensions.In order to facilitate the model, some rational assumptions for networked control systems are introduced as follows.(A1)
Single data package is transmitted. The packet loss and sequence disorder are not taken into consideration during the transmission process.(A2)
Uncertain time delay exists during the data transmission process. But time delay is bounded, and the maximum time delay does not exceed one sampling period; namely,τ∈[0,T], where T is the sampling period.(A3)
Sensor is clock driving; controller and actuator are all event driving.Based on the above assumptions (see Figure2), within a sampling period, the system input is not a constant value but is a piecewise constant. In a cycle, system input can be described as (2)ut=uk-1,tk<t≤tk+τkuk,tk+τk<t≤tk+T,where tk is the kth cycle sampling time and τk is the kth cycle delay.Figure 2
Timing diagram of signals transmitting in NCS.The discrete-time model of system (1) can be obtained as follows:(3)xk+1=Axk+B1τkuk+B2τkuk-1=Axk+B1τkuk+B-B1τkuk-1,where A=eAoT, B1(τk)=∫0T-τkeAotBodt, B2(τk)=∫T-τkTeAotBodt, and B=∫0TeAotBodt.Moreover, according to the Jordan canonical form, matrixA0 can be described as follows:(4)A0=Λdiag0p1×p1,J⌢p2×p2,J˘p3×p3Λ-1,where 0≤pi≤n (i=1,2,3), p1+p2+p3=n; J⌢ is Jordan block that is a diagonal matrix consisting of the different eigenvalues with a total of p2, denoted by λ1,λ2,…,λp2, while J˘ is Jordan block for the repeated eigenvalues with a total of p3, denoted by λ0; matrix Λ is the transformational matrix of Jordan canonical form for matrix A0.Correspondingly, the matrixB1 with the variable τk can be equivalently calculated as (5)B1τk=ΛdiagTs-τk,…,Ts-τk,eλ1Ts-τk-1λ1,…,eλp2Ts-τk-1λp2,JΛ-1B0,where(6)J=eλ0Ts-τk-1λ0∫0Ts-τkteλ0dt⋯∫0Ts-τkteλ0dtp3-1!0eλ0Ts-τk-1λ0⋯∫0Ts-τkteλ0dtp3-2!⋮⋮⋱⋮00⋯eλ0Ts-τk-1λ0p3×p3.Here a set of real numbers which are not equal to zero are denoted by α,α1,α2,…,αp2,α0.And we define(7)E=Λ-1B0Fτk=diagTs-τkα,…,Ts-τkα,eλ1Ts-τk-1α1λ1,…,eλp2Ts-τk-1αp2λp2,Jα0.From (7), we know appropriate values of α,α1,α2,…,αp2,α0 always can be chosen to satisfy FT(τk)F(τk)≤In×n.Comparing equality (7) with equality (5), we only need to define D=Λdiag(α,α1,α2,…,αp2,α0); the matrix B1(τk) can be expressed as (8)B1τk=DFτkE.Submitting equality (8) into equality (3), the following model can be obtained: (9)xk+1=Axk+DFτkEuk+B-DFτkEuk-1.Assume that the system is fully measurable; state feedback is introduced as follows:(10)uk=Kxk.Considering the sensor faults that may occur, the controller is expanded as (11)uFk=KGkxk,where uF(k)=[u1F(k),u2F(k),…,umF(k)]T represents faulty signal. G(k)=diag(g1(k),g2(k),…,gm(k)); gi=0 represents the fact that sensor i faults occur; gi=1 represents the fact that sensor i is normal; 0≤gi≤1 represents the fact that partial faults occur at sensor i. When G=I, it represents the fact that all sensors are normal. The situation that all actuators’ failure occurs at the same time is not taken into consideration here.Moreover, the boundaries of faults usually can be measured in the practical work. The following definition is presented.Definition 1.
The upper bound of fault matrices is defined as follows:(12)Gu=diaggu1,gu2,…,gun,1≥gui>0,while the lower bound of fault matrices is defined as follows:(13)Gl=diaggl1,gl2,…,gln,1>gli≥0.That is to say, G(k)∈[Gl,Gu], which is time-varying. The mean value of matrices in Definition 1 can also be obtained as (14)G0=diagg01,g02,…,g0n,g0i=gui+gli2;furthermore, the following matrices are introduced:(15)Lk=diagl1k,l2k,…,lnk,lik=gik-g0ig0i.Obviously, we have(16)-1≤gli-g0ig0i≤lik=gik-g0ig0i≤gui-g0ig0i=gui-gligui+gli≤1.Based on (16), we have -In×n≤L(k)≤In×n. Based on (16), the following can be obtained: (17)gi=g0i1+li,i=1,2,…n;Naturally,wehaveG=G0I+L.The model of closed-loop systems with sensor faults can be obtained as(18)xk+1=Axk+DFτkEKG0I+Lkxk+B-DFτkEKG0I+Lk-1xk-1.Remark 2.
The NCS with time-varying delay and sensor faults is modeled as a closed-loop system (18) with the uncertain parameter F(τk) and time-varying parameter L(k); unlike the previous models as [5], this model is related to the boundary values of the faults Gu and Gl. Moreover, according to the expression of matrix G0, we undoubtedly know it is invertible.


## 3. The Design of Guaranteed Cost Fault-Tolerant Control

For the system model (18) established above, the cost function is given as follows:(19)J∞=∑k=0∞xTkQxk+KG0I+LxkT×RKG0I+LxkKG0I+LxkT,where Q and R are symmetric positive definite matrices.To analyze the stability of the system expediently, the following lemmas are introduced.Lemma 3 (Schur complement).
For a symmetric matrixS=S11S12S21S22, where S11=S11T, S12T=S21, and S22=S22T, the following three conditions are equivalent:(1)
S<0;(2)
S11<0, S22-S12TS11-1S12<0;(3)
S22<0, S11-S12S22-1S12T<0.Lemma 4 (see [11]).
For any matricesW, M, N, or F(t) with FTF≤I and any scalar ε>0, the following inequality holds. W+MF(t)N+NTFT(t)MT≤W+εMMT+ε-1NTN.Theorem 5.
Given symmetric positive definite matricesQ, R and gain matrix K, if symmetric positive definite matrices P and S exist, as well as a scalar ε>0, satisfying(20)εDDT-P-1ABKG0I+L0∗-P+S+KG0I+LTRLKG0I+L+Q0EKG0I+LT∗∗-S-EKG0I+LT∗∗∗-εI<0,
then the NCS (18) is asymptotically stable, where ∗ represents the symmetry blocks of matrix.Proof.
Consider the following Lyapunov function:(21)Vk=xTkPxk+xTk-1Sxk-1.For the convenience of writing, we denote L=L(k) in the following expressions. Based on (18) conducting subtraction calculation can be obtained(22)ΔVk=vk+1-vk=xTk+1Pxk+1+xTkSxk-xTkPxk-xTk-1Sxk-1=Axk+DFEKG0I+Lxk+B-DFEKG0I+Lxk-1T×PAxk+DFEKG0I+Lxk+B-DFEKG0I+Lxk-1+xTkSxk-xTkPxk-xTk-1Sxk-1=xkxk-1TΓTPΓ-P+SΓTPΩ∗ΩTPΩ-S×xkxk-1,where Γ=A+DFEKG0(I+L) and Ω=BKG0(I+L)-DFEKG0(I+L).
System (18) is asymptotically stable, only if it satisfies ΔV<0. It is equivalent to(23)ΓTPΓ-P+SΓTPΩ∗ΩTPΩ-S<0.Now, take the following inequality into consideration:(24)ΓTPΓ-P+S+Q+KG0I+LTRKG0I+L∗ΓTPΩΩTPΩ-S<0.
Inequality (24) is equivalent to(25)ΓTPΓ-P+S∗ΓTPΩΩTPΩ-S+Q+KG0I+LTRLKG0I+L000<0.Q and R are symmetric positive definite matrices; therefore, inequality (24) is a sufficient condition of inequality (23). Inequality (24) can be expressed as follows: (26)-P+S+Q+KG0I+LTRKG0I+L∗0-S+ΓTΩTPΓΩ<0.From Lemma 3, it follows that(27)-P-1ΓΩ∗-P+S+Q+KG0I+LTRKG0I+L0∗∗-S<0.Inequality (27) can be rewritten as(28)-P-1ABKG0I+L∗-P+S+Q+KG0I+LTRKG0I+L0∗∗-S+D00F0EKG0I+L-EKG0I+L+0EKG0I+L-EKG0I+LTFTD00T<0.
From Lemma4, the sufficient condition of inequality (28) follows that(29)-P-1ABKG0I+L∗-P+S+Q+KG0I+LTRKG0I+L0∗∗-S+ε-10EKG0I+LT-EKG0I+LT0EKG0I+LT-EKG0I+LTT+εD00DT00<0.
Therefore, inequality (29) is a sufficient condition of inequality (23). That is to say, if inequality (29) exists, NCS (18) is asymptotically stable. From Lemma 3, the inequality above is equivalent to inequality (20). This completes the proof.Theorem 6.
If the conditions of Theorem5 are satisfied, for all allowable uncertainties of system (18), its cost function is defined as (19) satisfying (30)J∞≤J0=xT0Px0+xT-1Sx-1.Proof.
If inequality (20) exists, inequality (25) must exist.
Pre- and postmultiplying (25) by x(k)x(k-1)Tand by x(k)x(k-1) separately, it follows that(31)xkxk-1TΓTPΓ-P+SΓTPΩ∗ΩTPΩ-Sxkxk-1≤xkxk-1T-Q-KG0I+LTRKG0I+L000×xkxk-1=xTk-Q-KG0I+LTRKG0I+Lxk.Therefore,(32)J∞≤-∑k=0∞ΔVk=V0-limk→∞Vk=xT0Px0+xT-1Sx-1-limk→∞Vk.According to Theorem 5, system (18) is asymptotically stable. Therefore, limk→∞x(k)=limk→∞x(k-1)=0. Logically, there must exist limk→∞V(k)=0. Inserting it into (32), inequality (30) can be obtained.Theorem 7.
Given symmetric positive definite matricesQ and R, as well as a set of constants σ1 and σ2, if symmetric positive definite matrices X, Y and matrix W exist, as well as scalar ε>0, satisfying the following LMI:(33)-Q-1000000X00∗-σ1I00000X00∗∗-σ2I00000X0∗∗∗-σ1X0WT000EWT∗∗∗∗-σ2X0BWT00-EWT∗∗∗∗∗-R-10W00∗∗∗∗∗∗εDDT-XABW0∗∗∗∗∗∗∗-X+Y0EWT∗∗∗∗∗∗∗∗-Y-EWT∗∗∗∗∗∗∗∗∗-εI<0,
then the NCS (18) is asymptotically stable with control gain K=WX-1G0-1, and for all allowable uncertainties of system, its performance indicator defined as (19) satisfies inequality (30).Proof.
If inequality (20) exists, Theorems 5 and 6 must exist. Inequality (20) is equivalent to(34)εDDT-P-1ABKG0I+L0∗-P+S+Q0EKG0I+LT∗∗-S-EKG0I+LT∗∗∗-εI+0KG0I+LT00R0KG0I+L00<0.
From Lemma3, it follows that(35)-R-10KG0I+L0∗εDDT-P-1ABKG0I+L∗∗-P+S+Q0∗∗∗-P∗∗∗∗00EKG0I+LT-EKG0I+LT-εI<0.
Inequality (35) can be rewritten as (36)Φ+Ξ1LI^1+Ξ2LI^2+Ξ1LI^1T+Ξ2LI^2T<0,where(37)Φ=-R-10KG00∗εDDT-P-1ABKG0∗∗-P+S+Q0∗∗∗-P∗∗∗∗00EKG0T-EKG0T-εI<0,Ξ1=KG0000EKG0,I^1=00I00T,Ξ2=0BKG000-EKG0,I^1=000I0T.
Using Lemma4, it can be obtained that(38)Φ+σ1-1Ξ1Ξ1T+σ1I^1TI^1+σ2-1Ξ2Ξ2T+σ2I^2TI^2<0.Using Lemma 3 repeatedly, the following inequality can be obtained:(39)-Q-1000000I00∗-σ1-1I00000I00∗∗-σ2-1I00000I0∗∗∗-σ1I0KG0T000EKG0T∗∗∗∗-σ2I0BKG0T00-EKG0T∗∗∗∗∗-R-10KG000∗∗∗∗∗∗εDDT-P-1ABKG00∗∗∗∗∗∗∗-P+S0EKG0T∗∗∗∗∗∗∗∗-S-EKG0T∗∗∗∗∗∗∗∗∗-εI<0.
Pre- and postmultiplying (39) by block-diag(I,σ1I,σ2I,P-1,P-1,I,I,P-1,P-1,I) and then letting X=P-1, W=KG0P-1=KG0X, Y=P-1SP-1, inequality (33) can be obtained. Because G0 is invertible, K can be obtained by calculating K=WX-1G0-1. This completes the proof.


## 4. Simulations

Consider the model of inverted pendulum device as follows:(40)x˙t=-0.5-0.51000-1.3000.10.2-0.300000.3xt+0.110.51.2ut.Consider the sampling period T=0.1 s; the network induced time delay satisfies τk<T and is time-varying. We choose the parameters as α1=1.3, α2=1.8, α3=-1.1, and α4=1.5. Computing (5)–(8), we have(41)D=01.61-0.5772000-0.905401.3-0.8050.238800000,E=0.495-0.60091.21491.2,(42)Fτk=diage-0.30.1-τk-1-0.39,e-0.50.1-τk-1-0.9,e-1.30.1-τk-11.43,e0.30.1-τk-10.45.Obviously, FT(τk)F(τk)<I is satisfied.And the following parameters are given:(43)Q=5000008000001300000220,R=1800,σ1=σ2=109.Considering the sensor faults that may occur, the boundaries of faults are given in Table1.Table 1
The boundaries of sensor faults.
Upper bound
Lower bound
Mean value
Sensor faults matrixG
[
0.9
0
0
0
0
0.9
0
0
0
0
0.86
0
0
0
0
0.78
]
[
0.1
0
0
0
0
0.2
0
0
0
0
0.12
0
0
0
0
0.2
]
[
0.5
0
0
0
0
0.55
0
0
0
0
0.49
0
0
0
0
0.49
]By making use of LMI toolbox in MATLAB to solve the linear matrix inequality (33), guaranteed cost fault-tolerant controller parameters of NCS can be obtained:(44)K=WX-1G0-1=-0.0756-1.03-0.7531-1.9025.It assumes the initial state of the system is x0=x-1=[2-1.21.5-1]T; the comprehensive performance indicator can be obtained J0=16.3685. The time-varying transmission delays produced in network are shown in Figure 3. When all sensors are normal, namely, G=I, the state responses of NCS are shown in Figure 4, from which we know the system gets steady at 40 s. When sensor faults vary in the scope of boundary values in Table 1, the state responses of NCS are shown in Figure 5, from which we can see the transition time of state response obviously becomes longer than that in Figure 4 because of the effects of time-varying sensor faults, but the system is still asymptotically stable and gets steady at 80 s. So, the performance of NCS can be well maintained by the guaranteed cost fault-tolerant controller, which demonstrates the effectiveness and feasibility of the approach proposed in this paper.Figure 3
The distribution of delays in NCS.Figure 4
State response of NCS without sensor faults.Figure 5
State response of NCS with time-varying sensor faults.


## 5. Conclusions

When time-varying sensor faults occur, guaranteed cost fault-tolerant control problem of networked control systems is studied in this paper. Using Lyapunov stability theory and linear matrix inequality (LMI) approach, a sufficient condition is given, which can render the closed-loop NCS with sensor faults asymptotically stable and can guarantee it to meet the requirements of performance indicator. And based on LMI, the method of designing guaranteed cost fault-tolerant controller is proposed. Moreover, the feasibility and effectiveness of this method have been demonstrated by a simulation example. The next research task will be analyzing the time-varying actuator faults to improve the performance of NCS further.



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