# Robust Fault-Tolerant Control for Uncertain Networked Control Systems with State-Delay and Random Data Packet Dropout

Authors: Xiaomei Qi; Chengjin Zhang; Jason Gu
Journal: Journal of Control Science and Engineering
 (2012)

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## Abstract

A robust fault-tolerant controller design problem for networked control system (NCS) with
random packet dropout in both sensor-to-controller link and controller-to-actuator link is investigated. A novel stochastic NCS model with state-delay, model uncertainty, disturbance, probabilistic sensor failure, and actuator failure is proposed. The random packet dropout, sensor failures, and actuator failures are characterized by a binary random variable. The sufficient condition for asymptotical mean-square stability of NCS is derived and the closed-loop NCS satisfiesH∞ performance constraints caused by the random packet dropout and disturbance. The fault-tolerant controller is designed by solving a linear matrix inequality. A numerical example is presented to illustrate the effectiveness of the proposed method.


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## Body

## 1. Introduction

Networked control systems (NCSs) are one type of distributed control systems, in which the information of control system components (sensors, controllers, actuators, etc.) is exchanged via communication networks (see Figure1). Compared with the conventional point-to-point control systems, NCSs have many advantages, such as low cost of installation and maintenance, ease of diagnosis, and flexible architectures. However, the network in the control systems also bring many problems, such as network-induced delay, packet dropout, multiple channel transmission (as in [1–4]). Recent researches have a deep look into the controller and filter design for NCSs without faults (see [5–10]) and the references therein.Figure 1
Schematic description of a distributed NCS.Actually, NCSs are more vulnerable to faults than conventional control systems, due to the complexity introduced by the network. It is very significant to guarantee security and reliability of NCSs, because modern technological systems rely on sophisticated control systems to meet increased performance and safety requirement. Therefore, research on fault-tolerant control (FTC) for NCSs has attracted more and more attention from both industry and academia.However, research on FTC for NCSs is quite different from the ones for conventional control systems in many aspects (see [11–15]). A suitable architecture for FTC of NCSs must take into consideration the dynamical behaviour of network. In most research on FTC for NCSs, the fault model is described in a static way (as in [16–20]). Actually, faults often happen in a random way, so it is suitable to be studied in a dynamic way (as in [21–25]). In [21], the probabilistic sensor reductions are modeled by using a random variable that obeys a specific distribution in a known interval. In [22], the entire sensor failures or missing measurements have been described as a Bernoulli distributed variable. The reliable control design is considered for NCSs against probabilistic actuator fault with different failure rates in [23]. But, only actuator failures or sensor failures are considered in [16–23].As we know, many engineering control systems such as conventional oil-chemical industrial processes, nuclear reactors, long transmission lines in pneumatic, hydraulic, and rolling mill systems, NCSs contain some time-delay effects, model uncertainties (as in [26, 27]), and external disturbances. However, most of the aforementioned researches discuss FTC for NCSs without model uncertainty and disturbance(as in [16–20, 23, 24]).Therefore, from the above description, considering random packet dropout, the robust FTC for state-delay uncertain NCSs with both probabilistic sensors failures and actuators failures is still a challenging problem.In this paper, we study the robust FTC problem for NCSs with random packet dropout in both sensor-to-controller (S-C) link and controller-to-actuator (C-A) link. A new stochastic NCS model with fault is proposed, which includes the state-delay, model uncertainty, disturbance, random packet dropout, probabilistic sensors failures, and actuators failures. The random packet dropout, the sensor failure and the actuator failure are described as a binary random variable. The aim of this paper is to design a dynamic fault-tolerant controller for the NCS including packet dropouts, both sensor failures and actuator failures. The closed-loop NCS can be asymptotical mean-square stability and satisfies the performance constraint.The rest of the paper is organized as follows. The problem is formulated in Section2, a new stochastic NCS with probabilistic sensors failures and actuators failures is modelled. Section 3 presents the integral analysis of asymptotical mean-square stability for stochastic NCS with sensor and actuator faults. Section 4 designs a dynamic fault-tolerant controller. Section 5 gives a numerical example to demonstrate the effectiveness of the proposed method. Concluding remarks are made in Section 6.


## 2. Problem Formulation

Consider the following uncertain linear state-delay system:(1)xk+1=(A+ΔA)xk+Adxk-d+Buk+D0ωk,yk=C0xk,zk=Cxk+Cdxk-d+D1ωk,
where xk∈ℝn,yk∈ℝm,uk∈ℝp,zk∈ℝq denote the state, the sensor measurement, the control input, and the controlled output, respectively. ωk∈ℝr is the disturbance input belonging to l2[0,∞).A,Ad,B,C0,C,Cd,D0,D1, are known real constant matrices with appropriate dimensions. d>0 is a known delay. ΔA denotes the model uncertainty, which satisfies ΔA=HFkN,FkTFk<I,Fk represents an unknown real-valued time-varying matrix.Figure1 shows a typical feedback loop of NCS. Due to network congestion, traffic load balancing, or other unpredictable network behavior, the network-induced delay, data packet dropout, disorder may occur at the same time. In this paper, we focus on the data packet dropout phenomenon. Some assumptions in this paper are as follows.(1)
The sensor is clock-driven, the controller and the actuator are event-driven.(2)
Data packet dropouts occur in both S-C link and C-A link.(3)
Data are single-packet transmission with timestamp.(4)
Ignore the effects of quantization and asynchronous error in this paper.Remark 1.
A clock-driven sensor can send measurements to network periodically and is often used in real-time computing. The advantage of event-driven controller/actuator is that the controller/actuator will be updated as soon as the new data packet comes.Remark 2.
Taking the Internet as an illustration, the Transmission Control Protocol (TCP) is one of the core protocols of the Internet Protocol Suite. TCP is responsible for verifying the correct delivery of data from client to server. Data can be lost in the intermediate network. TCP adds support to detect errors or lost data and to trigger retransmission until the data is correctly and completely received. Thus, TCP is optimized for accurate delivery rather than timely delivery. Furthermore, for real-time feedback control, it is appropriate to discard the old data and transmit a new packet if it is available.Therefore, we assume if the total network-induced delayτk is larger than a sampling period, the output terminal will actively discard this packet, which means the network-induced delay problem can be considered as a packet-dropout problem. And the receiver with a buffer can rearrange the packets by reading the information of timestamp; in this way, the data disorder problem will be solved. Hence, we only focus on the data packet dropout issue in this paper.The binary random variableθk is an identically distributed (i.i.d.) process. θk=1 means that there is no packet dropout, and the sensors and the actuators are reliable; θk=0 means packet is lost, and the sensors and the actuators have failures. The probability distribution of θk is P{θk=0}=p,P{θk=1}=1-p, where p∈(0,1) indicates the sensor/actuator failure rate and the packet dropout rate.Considering the channel from the sensor to the controller, the sensor measurementyk will be updated to y̅k as follows:(2)y̅k=θkyk+(1-θk)y̅k-1.Considering the channel from the controller to the actuator, the control outputuk will be updated as follows:(3)uk=θkuk+(1-θk)uk-1.Remark 3.
From expressions (2) and (3), when θk=0 at time k, the sensor measurements and control information at time k are missing. The last available measurement y̅k-1 and controller output uk-1 stored in a buffer are utilized to substitute the missing data, which means at least one packet can be transmitted successfully in a sampling period.Since the random variableθk also represents sensor failures and actuator failures, the dynamic fault-tolerant controller is designed(4)x̂k+1=Afx̂k+Bfy̅k,uk=Cfx̂k.Defineηk=[xkT,x̂kT]T,dk=[uk-1T,y̅k-1T]T, the stochastic NCS with probabilistic sensor failures and actuator failures(5)ηk+1=Ãηk+(θk-p)Ã2ηk+ÃdZηk-d+(θk-p)B̃1dk+B̃2dk+D̃1ωk,zk=C̃ηk+CdZηk-d+D1ωk,
where(6)Ã=Ã1+H̃FkÑ=[ApBCfpBfC0Af]+[H0]Fk[N0],Ã2=[0BCfBfC00],Ãd=[Ad0],B̃1=[-B00-Bf],B̃2=[(1-p)B00(1-p)Bf],C̃=[C0],D̃1=[D00],Z=[I0].The aim of this paper is to design a dynamic fault-tolerant controller for the NCS (5), such that for all the possible data packet dropout and failures, the system (5) satisfies the following requirements.(Q1) The closed-loop NCS (5) is asymptotically mean-square stable.(Q2) Under the zero-initial condition, the outputzk satisfies
(7)∑k=0∞E{‖zk‖2}<γ2∑k=0∞E{‖ωk‖2}+δ2∑k=0∞E{‖dk‖2}
for all nonzero dk,ωk, where γ,δ>0 are the scalars we will design.


## 3. The Stability Analysis of NCS

In this section, the stability analysis for the NCS (5) is discussed.Lemma 4 (as in [28]).
LetW=WT<0,H,N be matrices of appropriate dimensions, with F satisfying FTF≤I, then W+HFN+NTFTHT<0 holds, if and only if there exists a ε>0 such that W+εHHT+ε-1NTN<0.Definition 5.
The NCS with faults given by (5) with ωk=0,dk=0, is asymptotically mean-square stable, if for any initial state, limk→∞𝔼{∥zk∥2}=0 holds.Theorem 6.
Givenγ>0,δ>0, the system (5) is asymptotically mean-square stable, and the output zk satisfies (7), if there exist matrices P=PT>0, and Q=QT>0 satisfying
(8)[-P0PÃPÃdPD̃10σPB̃1PB̃20*-IC̃CdD10000**-P00σÃ2TP00ZT***-Q00000****-γ2I0000*****-P000******-δ2I00*******-δ2I0********-Q-1]<0.Proof.
LetΘk=[ηkT,ηk-1T,…,ηk-dT]T, for all nonzero ηk, consider the Lyapunov function Vk(Θk)=V1k+V2k with
(9)V1k=ηkTPηk,V2k=∑i=k-dk-1ηiTZTQZηi.
The difference of Vk is
(10)ΔV1k=E{V1k(Θk+1)}-V1k(Θk)=ηkTÃTPÃηk+ηkTÃTPÃdZηk-d+ηkTÃTPB̃2dk+ηkTÃTPD̃1ωk+E{(θk-p)2}ηkTÃ2TPÃ2ηk+E{(θk-p)2}ηkTÃ2TPB̃1dk+ηk-dTZTÃdTPÃηk+ηk-dTZTÃdTPÃdZηk-d+ηk-dTZTÃdTPB̃2dk+ηk-dTZTÃdTPD̃1ωk+E{(θk-p)2}dkTB̃1TPÃ2ηk+E{(θk-p)2}dkTB̃1TPB̃1dk+dkTB̃2TPÃηk+dkTB̃2TPÃdZηk-d+dkTB̃2TPB̃2dk+dkTB̃2TPD̃1ωk+ωkTD̃1TPÃηk+ωkTD̃1TPÃdZηk-d+ωkTD̃1TPB̃2dk+ωkTD̃1TPD̃1ωk-ηkTPηkΔV2k=E{V2k(Θk+1)}-V2k(Θk)=ηkT(ZTQZ)ηk-ηk-dTZTQZηk-d.
For𝔼{θk-p}=0, and denote 𝔼{(θk-p)2}=σ2, we have
(11)ΔVk=[ηkZηk-ddkωk]T[Φ11Φ12Φ13Φ14*Φ22Φ23Φ24**Φ33Φ34***Φ44][ηkZηk-ddkωk]<ζkTΦijζk(i,j=1,…,4),
where ζk=[ηkT,ηk-dTZT,dkT,ωkT]T,
Φ11=ÃTPÃ+ZTQZ+σ2Ã2TPÃ2-P,
Φ12=ÃTPÃd,
Φ13=ÃTPB̃2+σ2Ã2TPB̃1,
Φ14=ÃTPD̃1,
Φ22=ÃdTPÃd-Q,
Φ23=ÃdTPB̃2,
Φ24=ÃdTPD̃1,
Φ33=B̃2TPB̃2+σ2B̃1TPB̃1,
Φ34=B̃2TPD̃1,
Φ44=D̃1TPD̃1,* is implicitly defined by the fact that the matrix is symmetric.
Whenωk=0,dk=0, (11) is rewritten as
(12)ΔVk,ωk=dk=0=[ÃTPÃ+ZTQZ+σ2Ã2TPÃ2-PÃTPÃdÃdTPÃÃdTPÃd-Q]=Π.
From (8) and the Schur complement theorem, Π<0 is arrived. Therefore, for all nonzero ηk, we have ΔVk<0, then the NCS (5) with sensor and actuator fault is asymptotically mean-square stable.
Next, For any nonzeroωk,dk, it follows from (5), (8), and (11) that
(13)E{V(Θk+1)}-E{V(Θk)}+E{zkTzk}-γ2E{ωkTωk}-δ2E{dkTdk}≤E{ζkTΦijζk+ηkTC̃TC̃ηk+ηkTC̃TCdZηk-d+ηkTC̃TD1ωk+ηk-dTZTCdTC̃ηk+ηk-dTZTCdTCdZηk-d+ηk-dTZTCdTD1ωk+ωkTD1TC̃ηk+ωkTD1TCdZηk-d+ωkTD1TD1ωk-γ2ωkTωk-δ2dkTdk}.
Then, we have(14)E{V(Θk+1)}-E{V(Θk)}+E{zkTzk}-γ2E{ωkTωk}-δ2E{dkTdk}<0.
Now, summing (14) from 0 to ∞ with respect to k yields
(15)∑k=0∞E{zkTzk}<γ2∑k=0∞E{ωkTωk}+δ2∑k=0∞E{dkTdk}+E{V0}-E{V∞}.
Since the system (5) is asymptotically mean-square stable, we can get that the following inequality:
(16)∑k=0∞E{zkTzk}<γ2∑k=0∞E{ωkTωk}+δ2∑k=0∞E{dkTdk}
holds under the zero initial condition. The proof is thus, complete.


## 4. RobustH∞ Controller Design

In this section, a theorem will be proposed to solve the controller design problem for stochastic state-delay NCS (5).Theorem 7.
Given a scalarγ>0,δ>0, the system (5) is asymptotically mean-square stable, and the controlled output zk satisfies the H∞ constraints (7), if there exist real scalars ε>0, and matrices S=ST>0,Q=QT>0, and R=RT>0, and real matrices W1,W2, and W3, such that the following inequality holds:
(17)[Ωij]<0,i,j=1,…,16,
where Ω11=-S-1,
Ω12=-I,
Ω14=A+pBW3,
Ω15=A,
Ω16=Ad,
Ω17=D0,
Ω1,10=-σB,
Ω1,12=(1-p)B,
Ω1,15=H,
Ω21=-I,
Ω22=-R,
Ω24=RA+pW2C0+W1,
Ω25=RA+pW2C0,
Ω26=RAd,
Ω27=RD0,
Ω2,10=-σRB,
Ω2,11=-σW2,
Ω2,12=(1-p)RB,
Ω2,13=(1-p)W2,
Ω2,15=RH,
Ω33=-I,
Ω34=C,
Ω35=C,
Ω36=Cd,
Ω37=D1,
Ω41=Ω14T,
Ω42=Ω24T,
Ω43=CT,
Ω44=-S,
Ω45=-S,
Ω48=σW3TBT,
Ω49=σW3TBTR+σC0TW2T,
Ω4,14=I,
Ω4,16=ϵNT,
Ω51=AT,
Ω52=Ω25T,
Ω53=CT,
Ω54=-S,
Ω55=-R,
Ω59=σC0TW2T,
Ω5,14=Q,
Ω5,16=ϵNT,
Ω61=AdT,
Ω61=AdTR,
Ω63=CdT,
Ω66=-Q,
Ω71=D0T,
Ω72=D0TR,
Ω73=D1T,
Ω77=-γ2I,
Ω84=Ω48T,
Ω88=-S,
Ω89=-S,
Ω94=Ω49T,
Ω95=Ω59T,
Ω99=-R,
Ω10,1=-σBT,
Ω10,2=-σBTR,
Ω10,10=Ω11,11=Ω12,12=Ω13,13=-δ2I,
Ω11,2=Ω2,11T,
Ω12,1=Ω12,1T,
Ω12,2=Ω2,12T,
Ω13,2=Ω2,13T,
Ω14,4=I,Ω14,5=QT,
Ω14,14=-I,
Ω15,1=HT,
Ω15,2=HTR,
Ω15,15=Ω16,16=-εI,
Ω16,4=Ω16,5=ϵN, the rest of matrix entries are zero.
The fault-tolerant controller parameters are(18)Af=X12-1[(W1-pRBW3)S-1(Y12T)-1],Bf=X12-1W2,Cf=W3S-1(Y12T)-1.Proof.
The system (5) is a parameter-dependent system. By Lemma 4, (8) is rewritten as(19)[-P0PÃ1PÃdPD̃10σPB̃1PB̃20PH̃0*-IC̃CdD1000000**-P00σÃ2TP00ZT0ϵÑT***-Q0000000****-γ2I000000*****-P00000******-δ2I0000*******-δ2I000********-Q-100*********-ϵI0**********-ϵI]<0.Next, partition P and P-1 as
(20)P=[RX12X12TX22],P-1=[S-1Y12Y12TY22].
Define
(21)T1=[S-1IY12T0],T2=[IR0X12T].
Obviously, we have PT1=T2,T1TPT1=T1TT2. Performing the congruence transformation diag{T1T,I,T1T,I,I,T1TI,I,T1T,I,I} to (19), we obtain the following:
(22)[Λi,j]<0,i,j=1,…,16,
where Λ11=-S-1,
Λ12=-I,Λ14=AS-1+pBCfY12T,
Λ15=A,Λ16=Ad,
Λ17=D0,
Λ1,10=-σB,
Λ1,12=(1-p)B,
Λ1,15=H,Λ21=-I,
Λ22=-R,Λ24=(RA+pX12BfC0)S-1+(pRBCf+X12Af)Y12T,
Λ25=RA+pX12BfC0,
Λ26=RAd,
Λ27=RD0,
Λ2,10=-σRB,
Λ2,11=-σX12Bf,
Λ2,12=(1-p)RB,
Λ2,13=(1-p)X12Bf,
Λ2,15=RH,
Λ33=-I,Λ34=CS-1,
Λ35=C,Λ36=Cd,Λ37=D1,
Λ41=Λ41T,Λ42=Λ42T,
Λ43=S-1CT,
Λ44=-S-1,Λ45=-I,Λ48=σY12CfTBT,
Λ49=σY12CfTBTR+σS-1C0TBfTX12T,
Λ4,14=S-1,
Λ4,16=ϵS-1NT,
Λ51=AT,
Λ52=Λ25T,Λ53=CT,
Λ54=-S,Λ55=-R,Λ59=σC0TBfTX12T,
Λ5,12=I,Λ5,16=ϵNT,
Λ61=AdT,
Λ62=AdTR,Λ63=CdT,
Λ66=-Q,Λ71=D0T,
Λ72=D0TR,Λ73=D1T,
Λ77=-γ2I,
Λ84=Λ48T,
Λ88=-S-1,Λ89=-I,Λ94=Λ49T,
Λ95=Λ59T,
Λ99=-R,Λ10,1=-σBT,Λ10,2=-σBTR,
Λ11,2=-σX12Bf,
Λ12,1=(1-p)BT,Λ12,2=(1-p)BTR,
Λ13,2=(1-p)X12Bf,Λ14,4=S-1,Λ14,5=I,Λ10,10=Λ11,11=Λ12,12=Λ13,13=-δ2I,Λ14,14=-Q-1,
Λ15,15=Λ16,16=-ϵI,Λ15,1=HT,Λ15,2=HTR,Λ16,4=ϵNS-1,Λ16,5=ϵN, the rest of matrix entries are zero.
Applying the congruence transformationdiag{I,I,I,S,I,I,I,S,I,I,I,I,I,Q,I,I} to (22) again, then (17) is achieved. Therefore, by Theorem 6, the desired result follows immediately.
Next, we will design the scalarsγ and δ by solving the optimization problem,
(23)min⁡ϵ,S,R,Q>0,W1,W2,W3γ,δ,subjectto(17).
Using the robust control toolbox, we will get the optimal value of γ and δ, and the proof is thus, complete.Remark 8.
From Theorems6 and 7, we know if (17) is feasible, I-RS-1=X12Y12T<0, hence, the square and nonsingular matrices X12 and Y12 can be always found (as in [29]). Then, the fault-tolerant controller parameters (18) are obtained.


## 5. Simulation Example

Consider the system (1) with parameters (as in [22]) as follows:(24)A=[-0.30-0.300.60.20.500.7],Ad=[-0.10000.10000.2],B=[012],D0=[0.500.2],H=[0.50.51],C0=[112],C=[0.100],Cd=[0.100],N=[0.100],D1=0.1.
Choose the same parameters as [22], packet loss rate is p=0.1, delay constant is d=3. With the method in [22], the optimal performance γ is 3.3339. Using the proposed method in this paper, the optimal performance γ is 1.3614, which means the smaller performance has been obtained.Next, let the packet loss rate and fault probabilityp=0.3, delay constant is d=2, under the initial condition x(0)=[-0.200.5]T,γ=1.6845, and δ=2.0753, the fault-tolerant controller parameters are designed by Theorem 7 as follows.(25)Af=[0.13572.4132-0.9861-2.1036-1.0003-0.0584-1.5372-0.4739-2.9271],Bf=[-0.06970.7264-0.0983],Cf=[-0.1016-0.07120.2598].From Figure2, it can be seen that state responses under the designed controller can stabilize the NCS with data packet loss, probabilistic actuator failures, and/or sensor failures, which can illustrate the effectiveness of the proposed method.Figure 2
State responses under the fault-tolerant controller.


## 6. Conclusion

Motivated by robust FTC problem over networks, a new stochastic NCS model with fault is addressed, which includes the state-delay, model uncertainty, disturbance, random packet dropout, probabilistic sensors failures, and actuators failures. The random packet dropout in both S-C link and C-A link, the sensor failure and the actuator failure are described as a binary random variable. The sufficient condition for asymptotical mean-square stability of the NCS has been derived and the closed-loop NCS satisfiesH∞ performance constraints. Finally, by solving a linear matrix inequality, the fault tolerant controller is designed.



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*Source: 734758-2012-05-27.xml*